[{"Name":"Binomial Expansion","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Binomial Expansion","Duration":"13m 1s","ChapterTopicVideoID":26091,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"We are beginning a new topic, the binomial theorem."},{"Start":"00:04.380 ","End":"00:11.710","Text":"This is usually due to famous Sir Isaac Newton."},{"Start":"00:11.980 ","End":"00:16.880","Text":"We\u0027ll start with something called binomial expansion."},{"Start":"00:16.880 ","End":"00:19.295","Text":"You\u0027ve seen this before."},{"Start":"00:19.295 ","End":"00:23.525","Text":"We want to evaluate a plus b to various powers."},{"Start":"00:23.525 ","End":"00:27.385","Text":"I\u0027ll start with 1 in the middle, a plus b^2."},{"Start":"00:27.385 ","End":"00:34.120","Text":"We know that this is equal to a^2 plus 2ab plus b^2."},{"Start":"00:35.950 ","End":"00:38.210","Text":"There\u0027s a few more,"},{"Start":"00:38.210 ","End":"00:40.460","Text":"a plus b to the power of 1."},{"Start":"00:40.460 ","End":"00:46.125","Text":"This is trivial, this is just a plus b itself."},{"Start":"00:46.125 ","End":"00:48.690","Text":"We\u0027d like to start from 0,"},{"Start":"00:48.690 ","End":"00:52.695","Text":"so a plus b^0 is just 1."},{"Start":"00:52.695 ","End":"00:55.055","Text":"How about a plus b^3?"},{"Start":"00:55.055 ","End":"00:57.175","Text":"You may or may not remember this,"},{"Start":"00:57.175 ","End":"01:01.480","Text":"but we can certainly derive it if we just multiply this by this."},{"Start":"01:01.480 ","End":"01:07.970","Text":"Anyway, the answer is a^3 plus 3a^b plus 3ab^2 plus b^3."},{"Start":"01:13.230 ","End":"01:20.370","Text":"In general, our main task is to find some formula for"},{"Start":"01:20.370 ","End":"01:26.920","Text":"a plus b raised to any power n. n is a positive whole number,"},{"Start":"01:26.920 ","End":"01:29.440","Text":"or at least non-negative."},{"Start":"01:29.440 ","End":"01:31.895","Text":"See if we can find a pattern,"},{"Start":"01:31.895 ","End":"01:34.240","Text":"a formula for doing this."},{"Start":"01:34.240 ","End":"01:37.240","Text":"Now there are patterns here."},{"Start":"01:37.240 ","End":"01:41.920","Text":"If we just study even these first 4 rows,"},{"Start":"01:41.920 ","End":"01:44.455","Text":"we can say various things."},{"Start":"01:44.455 ","End":"01:46.930","Text":"First of all, that if this is n,"},{"Start":"01:46.930 ","End":"01:48.344","Text":"and is 0,"},{"Start":"01:48.344 ","End":"01:49.640","Text":"1, 2, 3,"},{"Start":"01:49.640 ","End":"01:52.695","Text":"we can see that there are n plus 1 terms."},{"Start":"01:52.695 ","End":"01:57.054","Text":"For example, here with a plus b^2 there are 1,"},{"Start":"01:57.054 ","End":"01:58.540","Text":"2, 3 terms."},{"Start":"01:58.540 ","End":"02:01.900","Text":"Terms are what is separated by plus or minus."},{"Start":"02:01.900 ","End":"02:05.590","Text":"Here there\u0027s 1 term."},{"Start":"02:06.770 ","End":"02:09.780","Text":"1, here we have 2,"},{"Start":"02:09.780 ","End":"02:11.175","Text":"here we have 3,"},{"Start":"02:11.175 ","End":"02:13.170","Text":"here we have 4."},{"Start":"02:13.170 ","End":"02:15.890","Text":"If we write what n is,"},{"Start":"02:15.890 ","End":"02:19.820","Text":"n here is, which is the coefficient, is 0, 1, 2, 3,"},{"Start":"02:21.170 ","End":"02:26.810","Text":"and we want to generalize afterwards to a general n. But"},{"Start":"02:26.810 ","End":"02:32.315","Text":"we see that there are n plus 1 terms."},{"Start":"02:32.315 ","End":"02:38.570","Text":"We can also see that they all begin and end with a 1,"},{"Start":"02:38.570 ","End":"02:40.490","Text":"at least the coefficients do,"},{"Start":"02:40.490 ","End":"02:42.470","Text":"because I could put 1 here."},{"Start":"02:42.470 ","End":"02:44.030","Text":"We don\u0027t do that,"},{"Start":"02:44.030 ","End":"02:50.225","Text":"but they would all start and end in a 1 something."},{"Start":"02:50.225 ","End":"02:52.100","Text":"Notice that the powers,"},{"Start":"02:52.100 ","End":"02:54.080","Text":"well let\u0027s take this here."},{"Start":"02:54.080 ","End":"02:55.730","Text":"It\u0027s easier to see."},{"Start":"02:55.730 ","End":"02:57.905","Text":"The a\u0027s decrease."},{"Start":"02:57.905 ","End":"02:59.765","Text":"If we look here, the powers of a,"},{"Start":"02:59.765 ","End":"03:02.660","Text":"here we have a^2, here we have a,"},{"Start":"03:02.660 ","End":"03:04.390","Text":"which is a^1,"},{"Start":"03:04.390 ","End":"03:06.660","Text":"and here we don\u0027t have any a at all,"},{"Start":"03:06.660 ","End":"03:10.800","Text":"you could always imagine that it\u0027s like an a^0."},{"Start":"03:10.800 ","End":"03:13.620","Text":"Powers of a,"},{"Start":"03:13.620 ","End":"03:17.615","Text":"they decrease by 1 each time,"},{"Start":"03:17.615 ","End":"03:23.330","Text":"from n down to 0."},{"Start":"03:23.330 ","End":"03:27.050","Text":"Here if n is 2, we started off with a^2, a^1, a^0."},{"Start":"03:27.050 ","End":"03:28.730","Text":"Here a^3, a^2,"},{"Start":"03:28.730 ","End":"03:32.090","Text":"a^1, a^0, there is no a."},{"Start":"03:32.090 ","End":"03:42.330","Text":"The powers of b increase from 0 up to n. Here we have,"},{"Start":"03:42.330 ","End":"03:45.585","Text":"there\u0027s a b^0 here,"},{"Start":"03:45.585 ","End":"03:47.940","Text":"b^1, b^2,"},{"Start":"03:47.940 ","End":"03:52.805","Text":"b^3, and then here is 3."},{"Start":"03:52.805 ","End":"03:55.480","Text":"The pattern is that there\u0027s a coefficient,"},{"Start":"03:55.480 ","End":"04:00.310","Text":"and then powers of a that decrease and powers of b that increase."},{"Start":"04:00.310 ","End":"04:03.850","Text":"Notice also that if you add the powers of a and b,"},{"Start":"04:03.850 ","End":"04:07.585","Text":"like here if you add the 2 and the 1, you get 3,"},{"Start":"04:07.585 ","End":"04:09.950","Text":"and this is throughout,"},{"Start":"04:09.950 ","End":"04:11.755","Text":"3 plus 0 is 3,"},{"Start":"04:11.755 ","End":"04:17.080","Text":"2 plus 1 is 3, 1 plus 2 is 3, 0 plus 3 is 3."},{"Start":"04:17.080 ","End":"04:19.655","Text":"It\u0027s always a to the something,"},{"Start":"04:19.655 ","End":"04:21.765","Text":"call it k,"},{"Start":"04:21.765 ","End":"04:25.520","Text":"or let\u0027s make the power of b,"},{"Start":"04:25.520 ","End":"04:29.390","Text":"let\u0027s called that k. Here it will have n minus k."},{"Start":"04:29.390 ","End":"04:34.070","Text":"The total exponent is just n. This is k,"},{"Start":"04:34.070 ","End":"04:36.350","Text":"this will be n minus k. This is 2,"},{"Start":"04:36.350 ","End":"04:38.660","Text":"this will be 3 minus 2, so on."},{"Start":"04:38.660 ","End":"04:42.065","Text":"That\u0027s basically the pattern that we see,"},{"Start":"04:42.065 ","End":"04:45.270","Text":"and we have some coefficient in front."},{"Start":"04:45.270 ","End":"04:48.330","Text":"Each term is a coefficient, a to the something,"},{"Start":"04:48.330 ","End":"04:50.160","Text":"b to the something, the a is decreased,"},{"Start":"04:50.160 ","End":"04:51.505","Text":"the b is increased."},{"Start":"04:51.505 ","End":"04:54.780","Text":"Now let\u0027s try and formalize this,"},{"Start":"04:54.780 ","End":"04:56.975","Text":"or make it more precise."},{"Start":"04:56.975 ","End":"05:02.060","Text":"I\u0027m going to introduce something called Pascal\u0027s triangle."},{"Start":"05:02.060 ","End":"05:05.350","Text":"I don\u0027t want to lose this,"},{"Start":"05:05.350 ","End":"05:09.110","Text":"I copied it over here."},{"Start":"05:09.110 ","End":"05:12.210","Text":"Now we can scroll a bit."},{"Start":"05:12.520 ","End":"05:20.375","Text":"Let\u0027s look at this one just for example,"},{"Start":"05:20.375 ","End":"05:23.150","Text":"some number, a coefficient,"},{"Start":"05:23.150 ","End":"05:26.735","Text":"a power of a and a power of b."},{"Start":"05:26.735 ","End":"05:30.920","Text":"The sum of the powers together is n. In this case n is 3,"},{"Start":"05:30.920 ","End":"05:33.605","Text":"so it\u0027s a^2, b^1."},{"Start":"05:33.605 ","End":"05:38.990","Text":"What we\u0027re interested in is what is this co-efficient."},{"Start":"05:38.990 ","End":"05:42.290","Text":"Everything else can be predictably generated."},{"Start":"05:42.290 ","End":"05:47.660","Text":"I could immediately guess the next row as a^4,"},{"Start":"05:47.660 ","End":"05:50.084","Text":"then something, a^3,"},{"Start":"05:50.084 ","End":"05:51.140","Text":"b, and so on."},{"Start":"05:51.140 ","End":"05:53.780","Text":"But the numbers, the coefficients,"},{"Start":"05:53.780 ","End":"05:56.310","Text":"that\u0027s the hard part."},{"Start":"05:56.530 ","End":"06:00.140","Text":"We give them a term,"},{"Start":"06:00.140 ","End":"06:05.815","Text":"and this is a definition of this co-efficient here, we call it,"},{"Start":"06:05.815 ","End":"06:09.080","Text":"it\u0027s often pronounced n choose k,"},{"Start":"06:09.080 ","End":"06:11.539","Text":"because it comes from the world of combinatorics,"},{"Start":"06:11.539 ","End":"06:18.170","Text":"where we\u0027re choosing a certain number of items from a set of items."},{"Start":"06:18.170 ","End":"06:20.105","Text":"Anyway, don\u0027t want to get into the combinatorics."},{"Start":"06:20.105 ","End":"06:22.040","Text":"You can pronounce it n over k,"},{"Start":"06:22.040 ","End":"06:23.435","Text":"or just n k,"},{"Start":"06:23.435 ","End":"06:28.455","Text":"I\u0027ll say n choose k. That\u0027s the coefficient of"},{"Start":"06:28.455 ","End":"06:34.400","Text":"the term which has b^k,"},{"Start":"06:34.400 ","End":"06:36.829","Text":"where k is the lower number."},{"Start":"06:36.829 ","End":"06:40.940","Text":"It\u0027s in the expansion of a plus b^n,"},{"Start":"06:40.940 ","End":"06:44.300","Text":"so that the power of a would be n minus k,"},{"Start":"06:44.300 ","End":"06:47.360","Text":"because together they make up n. This number is n choose"},{"Start":"06:47.360 ","End":"06:51.940","Text":"k. There\u0027s something which is what I call off-by-one."},{"Start":"06:51.940 ","End":"06:55.550","Text":"There\u0027s a difference between the k,"},{"Start":"06:55.550 ","End":"06:59.880","Text":"which is the power of b, and the position."},{"Start":"06:59.920 ","End":"07:02.600","Text":"If you do computer science,"},{"Start":"07:02.600 ","End":"07:04.670","Text":"you might start counting from 0,"},{"Start":"07:04.670 ","End":"07:06.725","Text":"but if we count from 1,"},{"Start":"07:06.725 ","End":"07:12.620","Text":"then the first term has b^0 here."},{"Start":"07:12.620 ","End":"07:14.930","Text":"The second term has b^1,"},{"Start":"07:14.930 ","End":"07:17.550","Text":"the third term has b^2,"},{"Start":"07:17.550 ","End":"07:19.820","Text":"the fourth term has b^3."},{"Start":"07:19.820 ","End":"07:21.530","Text":"For the third term,"},{"Start":"07:21.530 ","End":"07:23.540","Text":"k is 2,"},{"Start":"07:23.540 ","End":"07:26.155","Text":"or if you specify k,"},{"Start":"07:26.155 ","End":"07:28.225","Text":"let\u0027s say is 4,"},{"Start":"07:28.225 ","End":"07:30.770","Text":"maybe not be in a future row,"},{"Start":"07:30.770 ","End":"07:33.035","Text":"then it\u0027s the 5th term."},{"Start":"07:33.035 ","End":"07:36.470","Text":"The k is 1 less than the term number,"},{"Start":"07:36.470 ","End":"07:39.740","Text":"because we start from k equals 0."},{"Start":"07:39.740 ","End":"07:44.000","Text":"This n choose k is the coefficient of,"},{"Start":"07:44.000 ","End":"07:46.010","Text":"instead of saying the coefficient of this,"},{"Start":"07:46.010 ","End":"07:49.715","Text":"the coefficient of the k plus first term;"},{"Start":"07:49.715 ","End":"07:52.765","Text":"the plus 1 reflects this off by 1."},{"Start":"07:52.765 ","End":"07:56.705","Text":"Or alternatively, if you want the kth term,"},{"Start":"07:56.705 ","End":"08:03.145","Text":"then the coefficient is n choose k minus 1."},{"Start":"08:03.145 ","End":"08:05.960","Text":"Now we\u0027ll give you examples in a moment."},{"Start":"08:05.960 ","End":"08:08.900","Text":"But each of these n choose k is 1 of these numbers,"},{"Start":"08:08.900 ","End":"08:10.364","Text":"1, 1, 1, 1,"},{"Start":"08:10.364 ","End":"08:12.380","Text":"2, 1, 1, 3, 3, 1."},{"Start":"08:12.380 ","End":"08:16.262","Text":"Also we\u0027re going to put these in a triangle and get Pascal\u0027s triangle,"},{"Start":"08:16.262 ","End":"08:19.860","Text":"so let\u0027s just get a bit more space here."},{"Start":"08:22.070 ","End":"08:24.820","Text":"There\u0027s one other thing before the triangles,"},{"Start":"08:24.820 ","End":"08:27.440","Text":"and that\u0027s the other notation."},{"Start":"08:27.540 ","End":"08:31.885","Text":"Instead of writing n choose k this way,"},{"Start":"08:31.885 ","End":"08:37.330","Text":"it\u0027s sometimes in the world of combinatorics written as a big C,"},{"Start":"08:37.330 ","End":"08:40.200","Text":"C for combinations actually,"},{"Start":"08:40.200 ","End":"08:44.070","Text":"and then the n here and the k here."},{"Start":"08:44.070 ","End":"08:50.409","Text":"Very importantly, there is a formula that computes these."},{"Start":"08:50.409 ","End":"08:53.200","Text":"If I want to know what is the value of n choose k,"},{"Start":"08:53.200 ","End":"08:55.180","Text":"you plug it in here."},{"Start":"08:55.180 ","End":"08:58.180","Text":"We\u0027ll see examples and how this relates."},{"Start":"08:58.180 ","End":"09:02.315","Text":"It will all tie together very shortly."},{"Start":"09:02.315 ","End":"09:05.025","Text":"That\u0027s the notation, that\u0027s the formula."},{"Start":"09:05.025 ","End":"09:06.855","Text":"Now let\u0027s see in practice."},{"Start":"09:06.855 ","End":"09:10.130","Text":"If you go back, we\u0027d gone up to here."},{"Start":"09:10.130 ","End":"09:14.460","Text":"We just take the coefficients without the a to the something,"},{"Start":"09:14.460 ","End":"09:16.025","Text":"b to the something,"},{"Start":"09:16.025 ","End":"09:19.175","Text":"what we had here is 1,"},{"Start":"09:19.175 ","End":"09:21.038","Text":"then 1, 1, then 1,"},{"Start":"09:21.038 ","End":"09:23.219","Text":"2, 1, then 1, 3, 3,"},{"Start":"09:23.219 ","End":"09:26.735","Text":"1, which is just like what\u0027s here."},{"Start":"09:26.735 ","End":"09:30.705","Text":"Let\u0027s just go back again, move on."},{"Start":"09:30.705 ","End":"09:34.970","Text":"In terms of this binomial coefficient symbol,"},{"Start":"09:34.970 ","End":"09:36.470","Text":"this 3 here,"},{"Start":"09:36.470 ","End":"09:40.525","Text":"for example, is 3 choose 1."},{"Start":"09:40.525 ","End":"09:43.620","Text":"Because we start from 0,"},{"Start":"09:43.620 ","End":"09:45.450","Text":"the k is what\u0027s underneath."},{"Start":"09:45.450 ","End":"09:48.075","Text":"This is a^2, b^1."},{"Start":"09:48.075 ","End":"09:50.010","Text":"b is to the power of 1,"},{"Start":"09:50.010 ","End":"09:53.205","Text":"so it\u0027s 3 choose 1."},{"Start":"09:53.205 ","End":"09:57.020","Text":"For example, this 1 would correspond to this."},{"Start":"09:57.020 ","End":"10:01.945","Text":"Here it\u0027s 1 choose 0 and 1 choose 1."},{"Start":"10:01.945 ","End":"10:09.055","Text":"Here remember it was a^2 plus 2ab plus b^2,"},{"Start":"10:09.055 ","End":"10:13.950","Text":"and there\u0027s a 1 here and a 1 here."},{"Start":"10:13.950 ","End":"10:16.260","Text":"This is this 1 here,"},{"Start":"10:16.260 ","End":"10:18.019","Text":"this is this 2 here,"},{"Start":"10:18.019 ","End":"10:20.724","Text":"and here it corresponds to 2,"},{"Start":"10:20.724 ","End":"10:22.340","Text":"2, and 2 at the top,"},{"Start":"10:22.340 ","End":"10:24.240","Text":"because it\u0027s all squared."},{"Start":"10:24.240 ","End":"10:26.224","Text":"The 0, 1,"},{"Start":"10:26.224 ","End":"10:32.705","Text":"2 is the exponent of b, b^0."},{"Start":"10:32.705 ","End":"10:36.430","Text":"We can add in here b^0."},{"Start":"10:36.430 ","End":"10:39.315","Text":"Well anyway, that\u0027s the 2 and that\u0027s the 2."},{"Start":"10:39.315 ","End":"10:41.835","Text":"Just study it for a while."},{"Start":"10:41.835 ","End":"10:46.130","Text":"These are the actual results of the coefficients."},{"Start":"10:46.130 ","End":"10:51.410","Text":"Here this is the same numbers but expressed in this notation."},{"Start":"10:51.410 ","End":"10:54.284","Text":"Now I haven\u0027t shown you that this formula works,"},{"Start":"10:54.284 ","End":"10:56.870","Text":"we\u0027ll see many examples of this later,"},{"Start":"10:56.870 ","End":"10:59.180","Text":"so far we\u0027ve just computed up to the third row."},{"Start":"10:59.180 ","End":"11:01.085","Text":"Let\u0027s take this 1 as an example."},{"Start":"11:01.085 ","End":"11:03.020","Text":"Let\u0027s see if we can get this 3."},{"Start":"11:03.020 ","End":"11:11.265","Text":"This 3 comes from the power of b^1 in the expansion of a plus b^3."},{"Start":"11:11.265 ","End":"11:13.830","Text":"This is what we want,"},{"Start":"11:13.830 ","End":"11:18.150","Text":"so 3 choose 1,"},{"Start":"11:18.150 ","End":"11:20.940","Text":"according to this formula, n is 3,"},{"Start":"11:20.940 ","End":"11:23.265","Text":"so it\u0027s 3 factorial,"},{"Start":"11:23.265 ","End":"11:24.870","Text":"k is 1,"},{"Start":"11:24.870 ","End":"11:27.690","Text":"so it\u0027s 1 factorial,"},{"Start":"11:27.690 ","End":"11:34.410","Text":"and n minus k is 3 minus 1, is 2 factorial."},{"Start":"11:34.410 ","End":"11:38.505","Text":"Again, this 2 is 3 minus 1 from here."},{"Start":"11:38.505 ","End":"11:43.110","Text":"Now 3 factorial is 6,"},{"Start":"11:43.110 ","End":"11:45.780","Text":"because it\u0027s 3 times 2 times 1, this is 6."},{"Start":"11:45.780 ","End":"11:47.835","Text":"1 factorial is 1,"},{"Start":"11:47.835 ","End":"11:49.890","Text":"2 factorial is 2,"},{"Start":"11:49.890 ","End":"11:53.670","Text":"6 over 1 times 2,"},{"Start":"11:53.670 ","End":"11:57.060","Text":"so this gives us equals 3,"},{"Start":"11:57.060 ","End":"12:03.200","Text":"and that\u0027s the 3 that\u0027s here, using this computation."},{"Start":"12:03.200 ","End":"12:06.425","Text":"Before we move on, just 1 other thing we should note,"},{"Start":"12:06.425 ","End":"12:10.670","Text":"is that there is a way of building this table"},{"Start":"12:10.670 ","End":"12:16.230","Text":"without computing a plus b to the power of 4,"},{"Start":"12:16.230 ","End":"12:17.665","Text":"then 5, then 6."},{"Start":"12:17.665 ","End":"12:20.330","Text":"Notice that if I have a certain row,"},{"Start":"12:20.330 ","End":"12:23.990","Text":"we\u0027ve got up to the row with 1, 3, 3, 1."},{"Start":"12:23.990 ","End":"12:29.225","Text":"If we add 3 plus 3 we get 6,"},{"Start":"12:29.225 ","End":"12:32.580","Text":"3 plus 1 we get 4."},{"Start":"12:32.750 ","End":"12:40.550","Text":"Here also. So each number in the following row is the sum of the 2 above it."},{"Start":"12:40.550 ","End":"12:42.290","Text":"1 plus 5 is 5,"},{"Start":"12:42.290 ","End":"12:44.225","Text":"4 plus 6 is 10,"},{"Start":"12:44.225 ","End":"12:46.475","Text":"6 plus 4 is 10, 4 plus 1 is 5."},{"Start":"12:46.475 ","End":"12:50.045","Text":"We always begin and end with a 1."},{"Start":"12:50.045 ","End":"12:52.130","Text":"Once we have a given row,"},{"Start":"12:52.130 ","End":"12:53.340","Text":"we put 1 at the beginning,"},{"Start":"12:53.340 ","End":"12:57.050","Text":"1 at the end, and all the other numbers of the sum of the 2 above."},{"Start":"12:57.050 ","End":"13:01.920","Text":"Let\u0027s take a break now and then continue with the binomial theorem."}],"Thumbnail":null,"ID":26918},{"Watched":false,"Name":"The Binomial Theorem","Duration":"17m 29s","ChapterTopicVideoID":26090,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.245","Text":"Continuing after the break, the binomial theorem."},{"Start":"00:04.245 ","End":"00:11.445","Text":"Now it\u0027s time to reveal what this theorem actually is, and here it is."},{"Start":"00:11.445 ","End":"00:15.030","Text":"It\u0027s a formula that says that when we take"},{"Start":"00:15.030 ","End":"00:19.395","Text":"the binomial a plus b and raise it to the power of n,"},{"Start":"00:19.395 ","End":"00:22.950","Text":"where n is a non-negative integer,"},{"Start":"00:22.950 ","End":"00:31.725","Text":"then it\u0027s equal to the sum of n choose k,"},{"Start":"00:31.725 ","End":"00:34.470","Text":"a^n minus k b^k."},{"Start":"00:34.470 ","End":"00:37.820","Text":"K goes from 0 to n. This is"},{"Start":"00:37.820 ","End":"00:43.295","Text":"the Sigma notation and in the examples we\u0027ll also write it out in full."},{"Start":"00:43.295 ","End":"00:53.360","Text":"The only thing that\u0027s missing here that we did earlier is the definition."},{"Start":"00:53.360 ","End":"00:55.640","Text":"We\u0027ve already covered it,"},{"Start":"00:55.640 ","End":"00:58.325","Text":"but let me refresh."},{"Start":"00:58.325 ","End":"01:01.440","Text":"N choose k,"},{"Start":"01:01.440 ","End":"01:03.850","Text":"or however you pronounce this symbol,"},{"Start":"01:03.850 ","End":"01:08.495","Text":"is equal to n factorial over"},{"Start":"01:08.495 ","End":"01:14.285","Text":"k factorial n minus k factorial."},{"Start":"01:14.285 ","End":"01:20.700","Text":"This together with this tells us how we expand a plus b to any power."},{"Start":"01:21.560 ","End":"01:24.715","Text":"Let\u0027s take n equals 4."},{"Start":"01:24.715 ","End":"01:28.580","Text":"We\u0027ve already done n goes from 0 to 3."},{"Start":"01:28.580 ","End":"01:32.375","Text":"If we go back, we did a plus b^3 in lower powers,"},{"Start":"01:32.375 ","End":"01:36.085","Text":"let\u0027s do a plus b^4 from the formula."},{"Start":"01:36.085 ","End":"01:44.605","Text":"First of all I\u0027ll use the Sigma to say that it\u0027s Sigma where k goes from 0 to 4."},{"Start":"01:44.605 ","End":"01:47.895","Text":"Remember this means it\u0027s going to be 5 terms,"},{"Start":"01:47.895 ","End":"01:49.560","Text":"0, 1, 2, 3,"},{"Start":"01:49.560 ","End":"01:55.170","Text":"4, there\u0027s always 1 more that\u0027s this of Pi 1 keeps recurring."},{"Start":"01:55.170 ","End":"02:04.660","Text":"N here is 4 choose k. K is the index that varies from 0 to 4."},{"Start":"02:06.620 ","End":"02:11.935","Text":"A^4 minus k, b^k."},{"Start":"02:11.935 ","End":"02:16.130","Text":"The powers of b are the ones that increase from 0 to 4."},{"Start":"02:16.130 ","End":"02:20.705","Text":"Next step is to open up the Sigma."},{"Start":"02:20.705 ","End":"02:22.445","Text":"We have 5 terms,"},{"Start":"02:22.445 ","End":"02:24.820","Text":"first 1, when k is 0,"},{"Start":"02:24.820 ","End":"02:27.280","Text":"4 choose 0,"},{"Start":"02:27.280 ","End":"02:30.455","Text":"a^4 minus 0 is 4,"},{"Start":"02:30.455 ","End":"02:33.760","Text":"and b^0, we don\u0027t write it."},{"Start":"02:33.760 ","End":"02:36.690","Text":"Always this comes out to be 1,"},{"Start":"02:36.690 ","End":"02:39.450","Text":"but at this stage we write it 4 choose 0."},{"Start":"02:39.450 ","End":"02:42.765","Text":"Next, 4 choose 1,"},{"Start":"02:42.765 ","End":"02:44.735","Text":"a^4 minus 1,"},{"Start":"02:44.735 ","End":"02:46.100","Text":"the a is decreasing,"},{"Start":"02:46.100 ","End":"02:47.180","Text":"the b is increasing."},{"Start":"02:47.180 ","End":"02:49.805","Text":"Here we have b^1 which is just b."},{"Start":"02:49.805 ","End":"02:52.225","Text":"Then 4 choose 2,"},{"Start":"02:52.225 ","End":"02:57.135","Text":"a this time is squared and b increases, squared."},{"Start":"02:57.135 ","End":"03:00.360","Text":"Then 4 choose 3,"},{"Start":"03:00.360 ","End":"03:02.820","Text":"a^2, so it\u0027s down to a."},{"Start":"03:02.820 ","End":"03:06.780","Text":"B increases to the power of 3."},{"Start":"03:06.780 ","End":"03:09.120","Text":"Then the last 1,"},{"Start":"03:09.120 ","End":"03:11.910","Text":"4 choose 4, a^0,"},{"Start":"03:11.910 ","End":"03:16.140","Text":"so just don\u0027t write it, and b^4."},{"Start":"03:16.140 ","End":"03:19.815","Text":"Of course, the first and the last 1,"},{"Start":"03:19.815 ","End":"03:22.565","Text":"that\u0027s always the case."},{"Start":"03:22.565 ","End":"03:27.610","Text":"Anyway, let\u0027s now compute these numbers."},{"Start":"03:27.610 ","End":"03:30.740","Text":"There\u0027s really 2 ways of going about this."},{"Start":"03:30.740 ","End":"03:36.695","Text":"1 is just computing straight from the formula or we could use Pascal\u0027s triangle."},{"Start":"03:36.695 ","End":"03:43.460","Text":"Since we got the triangle up to the 4th row actually,"},{"Start":"03:43.460 ","End":"03:44.630","Text":"but the 1 with 3 in it,"},{"Start":"03:44.630 ","End":"03:45.800","Text":"we had 1,"},{"Start":"03:45.800 ","End":"03:48.760","Text":"3, 3, 1."},{"Start":"03:48.760 ","End":"03:50.745","Text":"Above that we had 1,"},{"Start":"03:50.745 ","End":"03:54.445","Text":"2, 1 so on and so on."},{"Start":"03:54.445 ","End":"03:56.630","Text":"When we do the next row,"},{"Start":"03:56.630 ","End":"04:00.125","Text":"we start with a 1 and we end with a 1."},{"Start":"04:00.125 ","End":"04:03.860","Text":"Remember each term now is the sum of the 2 above it."},{"Start":"04:03.860 ","End":"04:06.245","Text":"So 1 and 3 is 4,"},{"Start":"04:06.245 ","End":"04:08.735","Text":"3 and 3 is 6,"},{"Start":"04:08.735 ","End":"04:11.915","Text":"3 and 1 is 4."},{"Start":"04:11.915 ","End":"04:16.610","Text":"The other way to compute these numbers,"},{"Start":"04:16.610 ","End":"04:23.950","Text":"is to use this formula for the binomial coefficient,"},{"Start":"04:23.950 ","End":"04:26.330","Text":"and we\u0027ll do it this way also,"},{"Start":"04:26.330 ","End":"04:28.910","Text":"but I won\u0027t bother with the first and the last,"},{"Start":"04:28.910 ","End":"04:31.640","Text":"it works when k is 0 and when k is n,"},{"Start":"04:31.640 ","End":"04:35.070","Text":"but we\u0027ll just do these 3."},{"Start":"04:35.090 ","End":"04:43.350","Text":"So 4 choose 1 is 4 factorial over 1 factorial,"},{"Start":"04:43.350 ","End":"04:50.085","Text":"and 4 minus 1 is 3 factorial, that\u0027s here."},{"Start":"04:50.085 ","End":"04:52.890","Text":"You know what, I\u0027ll write them and then we\u0027ll compute them."},{"Start":"04:52.890 ","End":"05:00.775","Text":"4 choose 2 is 4 factorial over 2 factorial and then 4 minus 2,"},{"Start":"05:00.775 ","End":"05:03.350","Text":"which is also 2 factorial,"},{"Start":"05:03.350 ","End":"05:08.660","Text":"and here 4 factorial over 3 factorial,"},{"Start":"05:08.660 ","End":"05:13.510","Text":"4 minus 3 is 1 factorial."},{"Start":"05:13.510 ","End":"05:17.310","Text":"Now 1 factorial is 1,"},{"Start":"05:17.310 ","End":"05:19.775","Text":"so it doesn\u0027t matter here and here."},{"Start":"05:19.775 ","End":"05:23.300","Text":"What we have 4 factorial is 24,"},{"Start":"05:23.300 ","End":"05:25.340","Text":"3 factorial is 6,"},{"Start":"05:25.340 ","End":"05:29.250","Text":"24 over 6 is 4."},{"Start":"05:29.250 ","End":"05:34.395","Text":"This 1 is 24 over 2 times 2,"},{"Start":"05:34.395 ","End":"05:37.435","Text":"24 over 4 is 6."},{"Start":"05:37.435 ","End":"05:40.535","Text":"This is the same as this just different order,"},{"Start":"05:40.535 ","End":"05:42.290","Text":"so it\u0027s also 4,"},{"Start":"05:42.290 ","End":"05:44.930","Text":"and that\u0027s what we got here, here and here."},{"Start":"05:44.930 ","End":"05:48.740","Text":"So Pascal\u0027s triangle method and"},{"Start":"05:48.740 ","End":"05:54.325","Text":"the formula method give us the same numbers and that\u0027s reassuring."},{"Start":"05:54.325 ","End":"05:56.925","Text":"I cleared some space here,"},{"Start":"05:56.925 ","End":"06:02.020","Text":"and now we can write the answer as a^4th,"},{"Start":"06:02.020 ","End":"06:04.335","Text":"the 1 we don\u0027t write,"},{"Start":"06:04.335 ","End":"06:06.375","Text":"and then from here,"},{"Start":"06:06.375 ","End":"06:09.570","Text":"we have 4a^3b,"},{"Start":"06:09.570 ","End":"06:12.510","Text":"4 over 2,"},{"Start":"06:12.510 ","End":"06:14.355","Text":"4 choose 2 is 6"},{"Start":"06:14.355 ","End":"06:22.740","Text":"a^2 b^2."},{"Start":"06:22.740 ","End":"06:25.905","Text":"Then again 4,"},{"Start":"06:25.905 ","End":"06:30.480","Text":"this times ab^3,"},{"Start":"06:30.480 ","End":"06:31.575","Text":"and the last 1,"},{"Start":"06:31.575 ","End":"06:37.500","Text":"1 if we don\u0027t write is just b^4."},{"Start":"06:37.500 ","End":"06:44.080","Text":"A plus b^4 is this."},{"Start":"06:44.270 ","End":"06:47.740","Text":"That\u0027s the example."},{"Start":"06:47.890 ","End":"06:54.350","Text":"Now let\u0027s look at some of the properties of the binomial coefficients,"},{"Start":"06:54.350 ","End":"06:58.985","Text":"and this also relates to Pascal\u0027s triangle."},{"Start":"06:58.985 ","End":"07:03.065","Text":"I\u0027ll show you 3 properties."},{"Start":"07:03.065 ","End":"07:11.255","Text":"The first property says n choose 0 is n choose n is 1."},{"Start":"07:11.255 ","End":"07:17.350","Text":"What this means is that the coefficients begin with 1 and end in 1."},{"Start":"07:17.350 ","End":"07:21.180","Text":"If n was like 4 here,"},{"Start":"07:21.180 ","End":"07:24.435","Text":"4 choose 0 is 1,"},{"Start":"07:24.435 ","End":"07:29.060","Text":"and 4 choose 4 is also 1."},{"Start":"07:29.060 ","End":"07:37.685","Text":"What this is saying is that these are all 1s and these are all 1s."},{"Start":"07:37.685 ","End":"07:40.595","Text":"That\u0027s the first property."},{"Start":"07:40.595 ","End":"07:45.250","Text":"The next property, I won\u0027t even read it out,"},{"Start":"07:45.250 ","End":"07:47.215","Text":"it\u0027s what is written."},{"Start":"07:47.215 ","End":"07:50.124","Text":"Essentially this is the symmetry property."},{"Start":"07:50.124 ","End":"07:56.125","Text":"Notice that if I put a vertical line through the middle, it\u0027s symmetrical."},{"Start":"07:56.125 ","End":"08:00.220","Text":"In our case, for example,"},{"Start":"08:00.220 ","End":"08:03.340","Text":"this came out to be 4,"},{"Start":"08:03.340 ","End":"08:08.000","Text":"and this also came out to be 4."},{"Start":"08:08.150 ","End":"08:14.730","Text":"The middle 1 is its own mirror image if it\u0027s an odd number of times the middle 1."},{"Start":"08:14.730 ","End":"08:18.150","Text":"But you notice that if you go from left to right,"},{"Start":"08:18.150 ","End":"08:20.310","Text":"1,4,6,4,1 or from right to left,"},{"Start":"08:20.310 ","End":"08:22.935","Text":"1,4,6,4,1 is the same."},{"Start":"08:22.935 ","End":"08:29.445","Text":"This says, in particular example I chose n is 4, k is 1."},{"Start":"08:29.445 ","End":"08:36.960","Text":"It says that 4 choose 1 is the same as full choose 4 minus 1, which is 3."},{"Start":"08:36.960 ","End":"08:40.110","Text":"In other words, if I subtract the 1 from the 4"},{"Start":"08:40.110 ","End":"08:44.340","Text":"and get 3 and put 3 here instead it\u0027s going to be the same."},{"Start":"08:44.340 ","End":"08:49.215","Text":"This is essentially what the symmetry is."},{"Start":"08:49.215 ","End":"08:51.779","Text":"I mean the entries,"},{"Start":"08:51.779 ","End":"08:57.030","Text":"the terms are symmetrical if this plus this is 4 or n in general."},{"Start":"08:57.030 ","End":"09:02.080","Text":"Notice that 0 plus 4 is 4,1 plus 3 is 4,"},{"Start":"09:02.080 ","End":"09:03.810","Text":"2 plus itself is 4."},{"Start":"09:03.810 ","End":"09:08.160","Text":"That\u0027s the symmetry condition and that\u0027s why it\u0027s n minus k together this,"},{"Start":"09:08.160 ","End":"09:09.525","Text":"and this adds up to n,"},{"Start":"09:09.525 ","End":"09:12.555","Text":"which means that they\u0027re symmetrically placed."},{"Start":"09:12.555 ","End":"09:16.620","Text":"As I said in the case of Pascal\u0027s triangle,"},{"Start":"09:16.620 ","End":"09:18.390","Text":"I mean you can look all over the place."},{"Start":"09:18.390 ","End":"09:21.746","Text":"5 is the same as 5,10 is the same as 10,"},{"Start":"09:21.746 ","End":"09:23.490","Text":"28 here, 28 here,"},{"Start":"09:23.490 ","End":"09:25.530","Text":"and so on and so on."},{"Start":"09:25.530 ","End":"09:32.250","Text":"The third property, this 1 won\u0027t even read it out just yet,"},{"Start":"09:32.250 ","End":"09:37.950","Text":"but this is the mathematical form of that property."},{"Start":"09:37.950 ","End":"09:39.660","Text":"Let\u0027s take an example."},{"Start":"09:39.660 ","End":"09:43.980","Text":"Let\u0027s say I put a triangle here."},{"Start":"09:43.980 ","End":"09:49.080","Text":"Notice that we said that each entry is the sum of the 2 above it."},{"Start":"09:49.080 ","End":"09:51.495","Text":"This is equal to this plus this."},{"Start":"09:51.495 ","End":"09:53.130","Text":"Well, what are these?"},{"Start":"09:53.130 ","End":"09:57.345","Text":"This 21 is in row 7."},{"Start":"09:57.345 ","End":"09:59.670","Text":"It\u0027s the eighth row if you count this row,"},{"Start":"09:59.670 ","End":"10:00.690","Text":"but when I say row 7,"},{"Start":"10:00.690 ","End":"10:02.850","Text":"I mean when the 7 is here."},{"Start":"10:02.850 ","End":"10:05.100","Text":"What this says is at 21,"},{"Start":"10:05.100 ","End":"10:07.635","Text":"which is 7,"},{"Start":"10:07.635 ","End":"10:14.280","Text":"and the number below is 1 more,"},{"Start":"10:14.280 ","End":"10:20.115","Text":"or rather 1 less than you count 1,2,3,"},{"Start":"10:20.115 ","End":"10:23.925","Text":"the third term, so the k is 2,"},{"Start":"10:23.925 ","End":"10:25.890","Text":"or you could start counting from 0."},{"Start":"10:25.890 ","End":"10:28.065","Text":"It\u0027s 1 way to do it, 0,1,2,"},{"Start":"10:28.065 ","End":"10:31.840","Text":"so that\u0027s 7,2, that\u0027s the 21."},{"Start":"10:32.000 ","End":"10:38.440","Text":"The 35 is the following term is 7."},{"Start":"10:38.600 ","End":"10:45.480","Text":"Choose 3, because k is 0,1,2,3."},{"Start":"10:45.480 ","End":"10:48.130","Text":"Well, it\u0027s just 1 more than the 2."},{"Start":"10:48.500 ","End":"10:57.570","Text":"The 56 is going to be the same index as the 35 because it\u0027s parallel."},{"Start":"10:57.570 ","End":"11:01.740","Text":"This is 0,1,2,3,0,1,2,3."},{"Start":"11:01.740 ","End":"11:08.175","Text":"That would be 8 choose 3 because we\u0027re on the row below,"},{"Start":"11:08.175 ","End":"11:11.640","Text":"which is, you can tell from the 8 here,"},{"Start":"11:11.640 ","End":"11:12.900","Text":"just like from the 7 here."},{"Start":"11:12.900 ","End":"11:14.355","Text":"It\u0027s 8 from here,"},{"Start":"11:14.355 ","End":"11:16.245","Text":"and then 0, 1, 2, 3,"},{"Start":"11:16.245 ","End":"11:18.000","Text":"or you count 1, 2, 3, 4,"},{"Start":"11:18.000 ","End":"11:19.960","Text":"but remember to subtract 1, so you get 3."},{"Start":"11:19.960 ","End":"11:23.775","Text":"That\u0027s 21, that\u0027s 35."},{"Start":"11:23.775 ","End":"11:27.135","Text":"This is 56."},{"Start":"11:27.135 ","End":"11:29.325","Text":"Just for extra practice,"},{"Start":"11:29.325 ","End":"11:30.795","Text":"let\u0027s compute them."},{"Start":"11:30.795 ","End":"11:33.480","Text":"7 choose 2."},{"Start":"11:33.480 ","End":"11:42.645","Text":"This would be 7 factorial/2 factorial 7 minus 2,"},{"Start":"11:42.645 ","End":"11:45.940","Text":"which is 5 factorial,"},{"Start":"11:49.850 ","End":"11:53.370","Text":"means you can compute it with the calculator,"},{"Start":"11:53.370 ","End":"11:57.390","Text":"but in these cases it\u0027s best to just multiply out 7 factorial is"},{"Start":"11:57.390 ","End":"12:03.135","Text":"7 times 6 times 5 times 4 times 3 times 2 times 1."},{"Start":"12:03.135 ","End":"12:13.380","Text":"Then 2 factorial is 2 times 1 and 5 factorial is 5 times 4 times 3 times 2 times 1."},{"Start":"12:13.380 ","End":"12:17.235","Text":"Notice that the 5 factorial and 5 factorial cancel."},{"Start":"12:17.235 ","End":"12:19.935","Text":"All we\u0027re left with is this bit here."},{"Start":"12:19.935 ","End":"12:22.230","Text":"7 times 6 over 2,"},{"Start":"12:22.230 ","End":"12:24.315","Text":"7 times 3,"},{"Start":"12:24.315 ","End":"12:27.670","Text":"that comes out to be 21."},{"Start":"12:27.670 ","End":"12:33.485","Text":"I will, leave you to check the formula that here we get 35,"},{"Start":"12:33.485 ","End":"12:37.080","Text":"then here we get 56."},{"Start":"12:37.220 ","End":"12:40.320","Text":"Yeah, I don\u0027t want to waste time doing that."},{"Start":"12:40.320 ","End":"12:43.105","Text":"It\u0027s just straightforward computation."},{"Start":"12:43.105 ","End":"12:47.780","Text":"Next, just 1 more skill we need."},{"Start":"12:47.780 ","End":"12:54.280","Text":"That is how to find a particular term of the binomial expansion."},{"Start":"12:54.280 ","End":"12:59.430","Text":"Just like before, we have to figure out a plus b^4."},{"Start":"12:59.430 ","End":"13:01.500","Text":"Sometimes we just want,"},{"Start":"13:01.500 ","End":"13:04.690","Text":"I don\u0027t know the third term."},{"Start":"13:07.480 ","End":"13:11.765","Text":"I wanted to remind you again of the off by 1 thing."},{"Start":"13:11.765 ","End":"13:18.240","Text":"Meaning that this binomial coefficient,"},{"Start":"13:18.240 ","End":"13:22.950","Text":"n choose k a to the n minus k b^k is not the kth term,"},{"Start":"13:22.950 ","End":"13:27.210","Text":"it\u0027s the k plus first term does there is this off by 1 thing."},{"Start":"13:27.210 ","End":"13:31.740","Text":"The exercise,"},{"Start":"13:31.740 ","End":"13:37.515","Text":"the example is find the sixth term in the expansion."},{"Start":"13:37.515 ","End":"13:40.125","Text":"Here I made it slightly different."},{"Start":"13:40.125 ","End":"13:42.720","Text":"Instead of a plus b got fed up of"},{"Start":"13:42.720 ","End":"13:46.770","Text":"a plus b to show you that it doesn\u0027t have to be those 2 letters."},{"Start":"13:46.770 ","End":"13:49.725","Text":"Let\u0027s do x plus y to the 15th."},{"Start":"13:49.725 ","End":"13:52.335","Text":"We want the sixth term."},{"Start":"13:52.335 ","End":"13:56.025","Text":"6 is 5 plus 1."},{"Start":"13:56.025 ","End":"14:01.680","Text":"What we\u0027re talking about is k equals 5 fixed,"},{"Start":"14:01.680 ","End":"14:04.740","Text":"meaning 0, 1, 2, 3, 4, 5."},{"Start":"14:04.740 ","End":"14:13.860","Text":"We count 6 starting from 0 and the n in this case is 15."},{"Start":"14:13.860 ","End":"14:21.345","Text":"What we want, the term will be 15,"},{"Start":"14:21.345 ","End":"14:27.450","Text":"choose 5, that\u0027s the n choose k. Then we\u0027ll want,"},{"Start":"14:27.450 ","End":"14:31.980","Text":"well, not a, but we\u0027ll have x instead."},{"Start":"14:31.980 ","End":"14:35.535","Text":"Of course, if you have different lattices, different here,"},{"Start":"14:35.535 ","End":"14:44.100","Text":"it\u0027s x to the power of n minus K. 15 minus 5 is 10."},{"Start":"14:44.100 ","End":"14:50.250","Text":"Here b, which is y^k, is this."},{"Start":"14:50.250 ","End":"14:52.125","Text":"This is what we want."},{"Start":"14:52.125 ","End":"14:58.380","Text":"The only thing missing is the actual computation of 15 choose 5."},{"Start":"14:58.380 ","End":"15:05.700","Text":"That would be 15 factorial/ 5 factorial."},{"Start":"15:05.700 ","End":"15:10.515","Text":"Then 15 minus 5 is 10 factorial."},{"Start":"15:10.515 ","End":"15:12.525","Text":"As we saw before,"},{"Start":"15:12.525 ","End":"15:14.190","Text":"whole chunk will cancel."},{"Start":"15:14.190 ","End":"15:15.945","Text":"If we write out the factors,"},{"Start":"15:15.945 ","End":"15:23.370","Text":"you can write 15 times 14 times 13 times 12 times 11 times."},{"Start":"15:23.370 ","End":"15:25.590","Text":"Notice what comes after the 11,"},{"Start":"15:25.590 ","End":"15:27.120","Text":"we get 10,"},{"Start":"15:27.120 ","End":"15:30.855","Text":"9,8, and so on,"},{"Start":"15:30.855 ","End":"15:33.405","Text":"right down to 1."},{"Start":"15:33.405 ","End":"15:37.980","Text":"On the denominator, we get 5 factorial,"},{"Start":"15:37.980 ","End":"15:43.215","Text":"which is 5 times 4 times 3 times 2 times 1."},{"Start":"15:43.215 ","End":"15:45.495","Text":"After that we get the term factorial,"},{"Start":"15:45.495 ","End":"15:49.695","Text":"which is 10,9,8,7,6 down to 1."},{"Start":"15:49.695 ","End":"15:55.020","Text":"This part cancels and we\u0027re left with just this."},{"Start":"15:55.020 ","End":"15:57.720","Text":"Of course we could do this on the calculator,"},{"Start":"15:57.720 ","End":"16:01.470","Text":"but let\u0027s see if we can cancel a bit, like Be creative."},{"Start":"16:01.470 ","End":"16:05.580","Text":"Yeah, 15 is 5 times 3."},{"Start":"16:05.580 ","End":"16:11.550","Text":"What else? 4 will go into"},{"Start":"16:11.550 ","End":"16:21.240","Text":"12,3 times and 2 will go into 14,7 times."},{"Start":"16:21.240 ","End":"16:25.730","Text":"What are we left with all the denominators gone and we\u0027re left with"},{"Start":"16:25.730 ","End":"16:33.750","Text":"7 times 13 times 3 times 11."},{"Start":"16:34.150 ","End":"16:43.135","Text":"I don\u0027t know, 7 times 13 is 91,3 times 11 is 33."},{"Start":"16:43.135 ","End":"16:46.110","Text":"I am going to need a calculator."},{"Start":"16:46.110 ","End":"16:51.525","Text":"It comes out to be 3,003."},{"Start":"16:51.525 ","End":"16:54.105","Text":"All we have to do is plug that here,"},{"Start":"16:54.105 ","End":"16:58.420","Text":"and the answer is,"},{"Start":"16:58.480 ","End":"17:00.590","Text":"well, I\u0027ll write it over here."},{"Start":"17:00.590 ","End":"17:08.970","Text":"3,003 times x^10 y^5."},{"Start":"17:09.730 ","End":"17:13.280","Text":"Highlight that."},{"Start":"17:13.280 ","End":"17:18.650","Text":"With this example that concludes the tutorial."},{"Start":"17:18.650 ","End":"17:24.875","Text":"There\u0027s a lot of solved exercises following."},{"Start":"17:24.875 ","End":"17:27.095","Text":"You should continue with those."},{"Start":"17:27.095 ","End":"17:29.490","Text":"Here we\u0027re done."}],"Thumbnail":null,"ID":26919},{"Watched":false,"Name":"Explanation and Examples - Part 1","Duration":"13m 1s","ChapterTopicVideoID":14883,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.115","Text":"Beginning a new topic;"},{"Start":"00:02.115 ","End":"00:04.185","Text":"the binomial theorem,"},{"Start":"00:04.185 ","End":"00:08.700","Text":"and this is usually due to Newton,"},{"Start":"00:08.700 ","End":"00:11.680","Text":"the famous Sir Isaac Newton."},{"Start":"00:11.980 ","End":"00:16.895","Text":"We\u0027ll start with something called binomial expansion,"},{"Start":"00:16.895 ","End":"00:19.310","Text":"you\u0027ve seen this before."},{"Start":"00:19.310 ","End":"00:23.530","Text":"We want to evaluate a plus b to various powers."},{"Start":"00:23.530 ","End":"00:27.305","Text":"I\u0027ll start with 1 in the middle, a plus b^2,"},{"Start":"00:27.305 ","End":"00:36.160","Text":"and we know that this is equal to a^2 plus 2ab plus b^2."},{"Start":"00:36.160 ","End":"00:38.210","Text":"There\u0027s a few more,"},{"Start":"00:38.210 ","End":"00:40.460","Text":"a plus b to the power of 1,"},{"Start":"00:40.460 ","End":"00:46.115","Text":"this is trivial, this is just a plus b itself."},{"Start":"00:46.115 ","End":"00:48.685","Text":"We\u0027d like to start from 0,"},{"Start":"00:48.685 ","End":"00:52.620","Text":"so a plus b to the 0 is just 1,"},{"Start":"00:52.620 ","End":"00:55.055","Text":"and how about a plus b cubed?"},{"Start":"00:55.055 ","End":"00:57.170","Text":"You may or may not remember this,"},{"Start":"00:57.170 ","End":"01:01.940","Text":"but we can certainly derive it if we just multiply this by this anyway,"},{"Start":"01:01.940 ","End":"01:07.980","Text":"the answer is a^3 plus 3a^2b plus 3ab^2 plus b^3."},{"Start":"01:13.250 ","End":"01:21.170","Text":"In general, our main task is to find some formula for a plus b"},{"Start":"01:21.170 ","End":"01:29.195","Text":"raised to any power n. N is a positive whole number or at least non-negative,"},{"Start":"01:29.195 ","End":"01:31.880","Text":"and to see if we can find a pattern,"},{"Start":"01:31.880 ","End":"01:34.250","Text":"a formula for doing this."},{"Start":"01:34.250 ","End":"01:37.210","Text":"Now, there are patterns here,"},{"Start":"01:37.210 ","End":"01:41.920","Text":"if we just study even these first 4 rows,"},{"Start":"01:41.920 ","End":"01:44.455","Text":"we can say various things."},{"Start":"01:44.455 ","End":"01:48.310","Text":"First of all, that if this is n and then 0, 1,"},{"Start":"01:48.310 ","End":"01:49.630","Text":"2, 3,"},{"Start":"01:49.630 ","End":"01:52.705","Text":"we can see that there are n plus 1 terms."},{"Start":"01:52.705 ","End":"01:55.780","Text":"For example, here with a plus b^2,"},{"Start":"01:55.780 ","End":"01:58.540","Text":"there are 1, 2, 3 terms."},{"Start":"01:58.540 ","End":"02:01.900","Text":"Terms are what are separated by plus or minus."},{"Start":"02:01.900 ","End":"02:05.590","Text":"Here there\u0027s 1 term,"},{"Start":"02:06.770 ","End":"02:10.080","Text":"1, here we have 2, here,"},{"Start":"02:10.080 ","End":"02:13.155","Text":"we have 3, here we have 4."},{"Start":"02:13.155 ","End":"02:16.810","Text":"If we write what n is, n here is,"},{"Start":"02:16.810 ","End":"02:20.737","Text":"which is the coefficient is 0,"},{"Start":"02:20.737 ","End":"02:22.000","Text":"1, 2, 3,"},{"Start":"02:22.000 ","End":"02:26.110","Text":"and we want to generalize afterwards to a general n,"},{"Start":"02:26.110 ","End":"02:32.310","Text":"but we see that there are n plus 1 terms."},{"Start":"02:32.310 ","End":"02:38.555","Text":"We can also see that they all begin and end with a 1,"},{"Start":"02:38.555 ","End":"02:42.470","Text":"at least the coefficients do because I could put 1 here."},{"Start":"02:42.470 ","End":"02:49.200","Text":"We don\u0027t do that, but they would all start and end in a 1 something."},{"Start":"02:49.360 ","End":"02:52.240","Text":"Notice that the powers,"},{"Start":"02:52.240 ","End":"02:54.090","Text":"well let\u0027s take this here,"},{"Start":"02:54.090 ","End":"02:57.905","Text":"its easier to see, the a\u0027s decrease."},{"Start":"02:57.905 ","End":"03:01.445","Text":"If we look here, the powers of a here we have a^2,"},{"Start":"03:01.445 ","End":"03:04.385","Text":"here we have a which is a^1,"},{"Start":"03:04.385 ","End":"03:06.740","Text":"and here we don\u0027t have an a at all,"},{"Start":"03:06.740 ","End":"03:11.480","Text":"it always imagine there\u0027s like an a^0."},{"Start":"03:11.480 ","End":"03:16.820","Text":"Powers of a decrease by 1 each"},{"Start":"03:16.820 ","End":"03:23.330","Text":"time from n down to 0."},{"Start":"03:23.330 ","End":"03:27.050","Text":"Here, if n is 2, we start off with a^2, a^1, a^0."},{"Start":"03:27.050 ","End":"03:29.105","Text":"Here, a^3, a^2,"},{"Start":"03:29.105 ","End":"03:32.090","Text":"a^1, a^0, there is no a."},{"Start":"03:32.090 ","End":"03:42.330","Text":"The powers of b increase from 0 up to n. Here we have,"},{"Start":"03:42.330 ","End":"03:47.505","Text":"b^0 here, b^1,"},{"Start":"03:47.505 ","End":"03:51.030","Text":"b^2, b^3,"},{"Start":"03:51.030 ","End":"03:52.815","Text":"and here is 3."},{"Start":"03:52.815 ","End":"03:55.650","Text":"The pattern is that there\u0027s a co-efficient,"},{"Start":"03:55.650 ","End":"03:58.320","Text":"and then powers of a that decrease,"},{"Start":"03:58.320 ","End":"04:00.295","Text":"and powers of b that increase."},{"Start":"04:00.295 ","End":"04:04.600","Text":"Notice also that if you add the powers of a and b like here,"},{"Start":"04:04.600 ","End":"04:07.575","Text":"if you add the 2 and the 1, you get 3,"},{"Start":"04:07.575 ","End":"04:09.645","Text":"and this is throughout,"},{"Start":"04:09.645 ","End":"04:12.170","Text":"like 3 plus 0 is 3,"},{"Start":"04:12.170 ","End":"04:14.530","Text":"2 plus 1 is 3,"},{"Start":"04:14.530 ","End":"04:17.040","Text":"1 plus 2 is 3, 0 plus 3 is3."},{"Start":"04:17.040 ","End":"04:19.655","Text":"Its always a to the something,"},{"Start":"04:19.655 ","End":"04:21.780","Text":"call it k,"},{"Start":"04:21.780 ","End":"04:25.530","Text":"or let\u0027s make the power of b,"},{"Start":"04:25.530 ","End":"04:26.580","Text":"lets call that k,"},{"Start":"04:26.580 ","End":"04:34.055","Text":"so here we\u0027ll have n minus k. The total exponent is just n. This is k,"},{"Start":"04:34.055 ","End":"04:36.340","Text":"this will be n minus k, This is 2,"},{"Start":"04:36.340 ","End":"04:38.450","Text":"this will be 3 minus 2 and so on."},{"Start":"04:38.450 ","End":"04:41.905","Text":"That\u0027s basically the pattern we see,"},{"Start":"04:41.905 ","End":"04:45.185","Text":"and we have some coefficient in front,"},{"Start":"04:45.185 ","End":"04:48.320","Text":"so each term is the coefficient a to the something,"},{"Start":"04:48.320 ","End":"04:51.485","Text":"b to the something, the a is decreased, the b is decreased."},{"Start":"04:51.485 ","End":"04:56.990","Text":"Now let\u0027s try and formalize this or make it more precise."},{"Start":"04:56.990 ","End":"05:02.060","Text":"I\u0027m going to introduce something called Pascal\u0027s triangle."},{"Start":"05:02.060 ","End":"05:05.030","Text":"I don\u0027t want to lose this."},{"Start":"05:05.030 ","End":"05:09.095","Text":"I copied it over here,"},{"Start":"05:09.095 ","End":"05:12.210","Text":"now we can scroll a bit."},{"Start":"05:12.520 ","End":"05:17.140","Text":"As we saw, each term is made up,"},{"Start":"05:17.140 ","End":"05:21.825","Text":"let\u0027s look at this one just for example, some number,"},{"Start":"05:21.825 ","End":"05:24.810","Text":"a coefficient, a power of a,"},{"Start":"05:24.810 ","End":"05:26.735","Text":"and a power of b."},{"Start":"05:26.735 ","End":"05:29.480","Text":"The sum of the powers together as n,"},{"Start":"05:29.480 ","End":"05:30.920","Text":"in this case n is 3,"},{"Start":"05:30.920 ","End":"05:33.605","Text":"so its a^2, b^1."},{"Start":"05:33.605 ","End":"05:37.085","Text":"What we\u0027re interested in is,"},{"Start":"05:37.085 ","End":"05:38.990","Text":"what is this co-efficient?"},{"Start":"05:38.990 ","End":"05:42.290","Text":"Everything else can be predictably generated,"},{"Start":"05:42.290 ","End":"05:47.660","Text":"I could immediately guess the next row as a^4,"},{"Start":"05:47.660 ","End":"05:51.140","Text":"then something a^3b and so on,"},{"Start":"05:51.140 ","End":"05:53.780","Text":"but the numbers, the coefficients,"},{"Start":"05:53.780 ","End":"05:56.310","Text":"that\u0027s the hard part."},{"Start":"05:56.540 ","End":"06:00.135","Text":"We give them a term,"},{"Start":"06:00.135 ","End":"06:03.860","Text":"and this is a definition that this co-efficient here,"},{"Start":"06:03.860 ","End":"06:05.830","Text":"we call it,"},{"Start":"06:05.830 ","End":"06:09.080","Text":"its often pronounced n choose k"},{"Start":"06:09.080 ","End":"06:11.840","Text":"because it comes from the world of combinatorics where we\u0027re"},{"Start":"06:11.840 ","End":"06:18.350","Text":"choosing number of items from a set of items anyway,"},{"Start":"06:18.350 ","End":"06:20.105","Text":"I don\u0027t want to get into the combinatorics."},{"Start":"06:20.105 ","End":"06:23.150","Text":"You can pronounce it n over k or just nk."},{"Start":"06:23.150 ","End":"06:28.470","Text":"I\u0027ll say n choose k. That\u0027s the coefficient of"},{"Start":"06:28.470 ","End":"06:34.115","Text":"the term which has b^k,"},{"Start":"06:34.115 ","End":"06:36.335","Text":"where k is the lower number,"},{"Start":"06:36.335 ","End":"06:40.915","Text":"and its in the expansion of a plus b^n,"},{"Start":"06:40.915 ","End":"06:45.810","Text":"so that the power of a would be n-k because together, they make up n,"},{"Start":"06:45.810 ","End":"06:52.525","Text":"so this number is n choose k. There\u0027s something which is what I call off by 1,"},{"Start":"06:52.525 ","End":"06:55.550","Text":"there\u0027s a difference between the k,"},{"Start":"06:55.550 ","End":"06:59.880","Text":"which is the power of b, the position."},{"Start":"06:59.920 ","End":"07:02.600","Text":"If you do computer science,"},{"Start":"07:02.600 ","End":"07:04.670","Text":"you might start counting from 0,"},{"Start":"07:04.670 ","End":"07:06.730","Text":"but if we count from 1,"},{"Start":"07:06.730 ","End":"07:12.615","Text":"then the first term has b^0 here."},{"Start":"07:12.615 ","End":"07:14.430","Text":"The second term has b^1,"},{"Start":"07:14.430 ","End":"07:17.430","Text":"the third term has b^2,"},{"Start":"07:17.430 ","End":"07:19.800","Text":"the fourth term has b^3."},{"Start":"07:19.800 ","End":"07:21.525","Text":"For the third term,"},{"Start":"07:21.525 ","End":"07:23.535","Text":"k is 2,"},{"Start":"07:23.535 ","End":"07:28.230","Text":"or if you specify k let\u0027s say as 4,"},{"Start":"07:28.230 ","End":"07:30.770","Text":"to maybe be in the future row,"},{"Start":"07:30.770 ","End":"07:33.020","Text":"then its the fifth term."},{"Start":"07:33.020 ","End":"07:39.740","Text":"The k is 1 less than the term number because we start from k equals 0."},{"Start":"07:39.740 ","End":"07:46.010","Text":"This n choose k is the coefficient of saying the coefficient of this,"},{"Start":"07:46.010 ","End":"07:49.715","Text":"the co-efficient of the k plus first term."},{"Start":"07:49.715 ","End":"07:52.765","Text":"The plus 1 reflects itself by 1."},{"Start":"07:52.765 ","End":"07:56.705","Text":"Or alternatively, if you want the kth term,"},{"Start":"07:56.705 ","End":"08:03.140","Text":"then the coefficient is n choose k minus 1."},{"Start":"08:03.140 ","End":"08:05.960","Text":"Now, I\u0027ll give you examples in a moment,"},{"Start":"08:05.960 ","End":"08:08.900","Text":"but each of these n choose k is 1 of these numbers,"},{"Start":"08:08.900 ","End":"08:10.365","Text":"1, 1, 1, 1,"},{"Start":"08:10.365 ","End":"08:12.680","Text":"2, 1, 1, 3, 3, 1."},{"Start":"08:12.680 ","End":"08:15.950","Text":"Also, we\u0027re going to put these in a triangle and get Pascal\u0027s triangle,"},{"Start":"08:15.950 ","End":"08:19.860","Text":"so let\u0027s just get a bit more space here."},{"Start":"08:22.070 ","End":"08:24.820","Text":"There\u0027s one other thing before the triangles,"},{"Start":"08:24.820 ","End":"08:27.440","Text":"and that\u0027s the other notation."},{"Start":"08:27.540 ","End":"08:31.780","Text":"Instead of writing n choose k this way,"},{"Start":"08:31.780 ","End":"08:37.330","Text":"its sometimes in the world of combinatorics written as a big C,"},{"Start":"08:37.330 ","End":"08:40.200","Text":"C for combinations actually,"},{"Start":"08:40.200 ","End":"08:44.055","Text":"and then the n here and the k here."},{"Start":"08:44.055 ","End":"08:50.190","Text":"Very importantly, there is a formula that computes these."},{"Start":"08:50.190 ","End":"08:53.230","Text":"If I want to know what is the value of n choose k,"},{"Start":"08:53.230 ","End":"08:55.195","Text":"you plug it in here,"},{"Start":"08:55.195 ","End":"08:58.180","Text":"we\u0027ll see examples and how this relates,"},{"Start":"08:58.180 ","End":"09:02.310","Text":"it\u0027ll tie together very shortly."},{"Start":"09:02.310 ","End":"09:05.025","Text":"That\u0027s notation, that\u0027s the formula,"},{"Start":"09:05.025 ","End":"09:06.845","Text":"and now let\u0027s see in practice,"},{"Start":"09:06.845 ","End":"09:09.980","Text":"if we go back, we got up to here."},{"Start":"09:09.980 ","End":"09:14.445","Text":"We just take the coefficients without the a to the something,"},{"Start":"09:14.445 ","End":"09:16.025","Text":"b to the something."},{"Start":"09:16.025 ","End":"09:19.185","Text":"What we had here is 1,"},{"Start":"09:19.185 ","End":"09:20.505","Text":"then 1 1,"},{"Start":"09:20.505 ","End":"09:21.660","Text":"1 2 1,"},{"Start":"09:21.660 ","End":"09:23.670","Text":"then 1 3 3 1,"},{"Start":"09:23.670 ","End":"09:26.610","Text":"which is just like what\u0027s here."},{"Start":"09:26.610 ","End":"09:30.410","Text":"Let\u0027s just go back again and move on,"},{"Start":"09:30.410 ","End":"09:34.970","Text":"and in terms of this binomial coefficient symbol,"},{"Start":"09:34.970 ","End":"09:37.880","Text":"this 3 here for example,"},{"Start":"09:37.880 ","End":"09:40.510","Text":"is 3 choose 1."},{"Start":"09:40.510 ","End":"09:43.620","Text":"Because we start from 0,"},{"Start":"09:43.620 ","End":"09:45.450","Text":"the k is what\u0027s underneath."},{"Start":"09:45.450 ","End":"09:49.830","Text":"This is a^2, b^1."},{"Start":"09:49.830 ","End":"09:53.190","Text":"b^1, so its 3 choose 1."},{"Start":"09:53.190 ","End":"09:57.045","Text":"For example, this 1 would correspond to this."},{"Start":"09:57.045 ","End":"09:58.635","Text":"Here its 1,"},{"Start":"09:58.635 ","End":"10:01.950","Text":"choose 0, and 1 choose 1."},{"Start":"10:01.950 ","End":"10:03.675","Text":"Here for the,"},{"Start":"10:03.675 ","End":"10:07.750","Text":"remember it was a^2 plus 2ab plus b^2,"},{"Start":"10:08.750 ","End":"10:12.820","Text":"and there\u0027s a 1 here and a 1 here."},{"Start":"10:13.400 ","End":"10:16.260","Text":"This is this 1 here,"},{"Start":"10:16.260 ","End":"10:17.820","Text":"this is this 2 here,"},{"Start":"10:17.820 ","End":"10:21.210","Text":"and here it corresponds to 2 2,"},{"Start":"10:21.210 ","End":"10:24.090","Text":"and 2 at the top because its all squared,"},{"Start":"10:24.090 ","End":"10:26.214","Text":"and the 0, 1,"},{"Start":"10:26.214 ","End":"10:32.715","Text":"2 is the exponent of b, b^0."},{"Start":"10:32.715 ","End":"10:36.930","Text":"We can add in here b^0."},{"Start":"10:36.930 ","End":"10:39.720","Text":"Well anyway that\u0027s the 2 and that\u0027s the 2,"},{"Start":"10:39.720 ","End":"10:41.835","Text":"just study it for awhile."},{"Start":"10:41.835 ","End":"10:45.870","Text":"These are the actual results of the coefficients,"},{"Start":"10:45.870 ","End":"10:48.130","Text":"and here this is the same numbers,"},{"Start":"10:48.130 ","End":"10:51.410","Text":"but expressed in this notation."},{"Start":"10:51.410 ","End":"10:54.470","Text":"Now I haven\u0027t shown you that this formula works."},{"Start":"10:54.470 ","End":"10:56.870","Text":"We\u0027ll see many examples of this later,"},{"Start":"10:56.870 ","End":"10:59.230","Text":"so far we\u0027ve just computed up to the third row,"},{"Start":"10:59.230 ","End":"11:01.085","Text":"let\u0027s take this 1 as an example."},{"Start":"11:01.085 ","End":"11:03.020","Text":"Let\u0027s see if we can get this 3."},{"Start":"11:03.020 ","End":"11:11.925","Text":"This 3 comes from the power of b^1 in the expansion of a plus b^3,"},{"Start":"11:11.925 ","End":"11:13.680","Text":"so this is what we want."},{"Start":"11:13.680 ","End":"11:19.920","Text":"3 choose 1 according to this formula,"},{"Start":"11:19.920 ","End":"11:23.265","Text":"n is 3, so its 3 factorial,"},{"Start":"11:23.265 ","End":"11:24.870","Text":"k is 1,"},{"Start":"11:24.870 ","End":"11:27.690","Text":"so it\u0027s 1 factorial,"},{"Start":"11:27.690 ","End":"11:34.410","Text":"and n minus k is 3 minus 1 is 2 factorial."},{"Start":"11:34.410 ","End":"11:38.490","Text":"Again, this 2 is 3 minus 1 from here."},{"Start":"11:38.490 ","End":"11:43.110","Text":"Now 3 factorial is 6,"},{"Start":"11:43.110 ","End":"11:45.830","Text":"because its 3 times 2 times 1 this is 6,"},{"Start":"11:45.830 ","End":"11:47.835","Text":"1 factorial is 1,"},{"Start":"11:47.835 ","End":"11:50.130","Text":"2 factorial is 2,"},{"Start":"11:50.130 ","End":"11:52.920","Text":"6 over 1 times 2,"},{"Start":"11:52.920 ","End":"11:57.060","Text":"so this gives us equals 3,"},{"Start":"11:57.060 ","End":"12:03.200","Text":"and that\u0027s the 3 that\u0027s here using this computation."},{"Start":"12:03.200 ","End":"12:09.590","Text":"Before we move on, just one other thing we should note is that there is a way of building"},{"Start":"12:09.590 ","End":"12:16.230","Text":"this table without computing a plus b to the power of 4,"},{"Start":"12:16.230 ","End":"12:17.665","Text":"then 5, then 6."},{"Start":"12:17.665 ","End":"12:23.990","Text":"Notice that if I have a certain row and we\u0027ve got up to the row with 1331,"},{"Start":"12:23.990 ","End":"12:31.884","Text":"If we add 3 plus 3 we get 6,"},{"Start":"12:31.884 ","End":"12:34.865","Text":"3 plus 1 we get 4, and here also,"},{"Start":"12:34.865 ","End":"12:41.285","Text":"so each number in the following row is the sum of the 2 above it."},{"Start":"12:41.285 ","End":"12:46.220","Text":"1 plus 4 is 5, 4 plus 6 is 10, 6 plus 4 is10,"},{"Start":"12:46.220 ","End":"12:50.045","Text":"4 plus 1 is 5, and we\u0027ll always begin and end with a 1."},{"Start":"12:50.045 ","End":"12:52.130","Text":"Once we have a given row,"},{"Start":"12:52.130 ","End":"12:53.340","Text":"we put one in the beginning,"},{"Start":"12:53.340 ","End":"12:57.050","Text":"one at the end, and all the other numbers of the sum of the 2 above."},{"Start":"12:57.050 ","End":"13:01.920","Text":"Let\u0027s take a break now and then continue with the binomial theorem."}],"Thumbnail":null,"ID":26920},{"Watched":false,"Name":"Explanation and Examples - Part 2","Duration":"17m 29s","ChapterTopicVideoID":14884,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.245","Text":"Continuing after the break, the Binomial Theorem."},{"Start":"00:04.245 ","End":"00:11.445","Text":"Now it\u0027s time to reveal what this theorem actually is, and here it is."},{"Start":"00:11.445 ","End":"00:15.030","Text":"It\u0027s a formula that says that when we take"},{"Start":"00:15.030 ","End":"00:19.395","Text":"the binomial a plus b and raise it to the power of n,"},{"Start":"00:19.395 ","End":"00:22.965","Text":"where n is a non-negative integer,"},{"Start":"00:22.965 ","End":"00:31.320","Text":"then it\u0027s equal to the sum of n, choose k,"},{"Start":"00:31.320 ","End":"00:34.170","Text":"a to the n minus k, b to the k,"},{"Start":"00:34.170 ","End":"00:39.620","Text":"and k goes from 0 to n. This is the Sigma notation,"},{"Start":"00:39.620 ","End":"00:43.295","Text":"and in the examples, we\u0027ll also write it out in full."},{"Start":"00:43.295 ","End":"00:52.935","Text":"The only thing that\u0027s missing here that we did earlier is the definition,"},{"Start":"00:52.935 ","End":"00:55.640","Text":"as I said we\u0027ve already covered it,"},{"Start":"00:55.640 ","End":"01:01.430","Text":"but let me refresh n choose k,"},{"Start":"01:01.430 ","End":"01:03.860","Text":"or however you pronounce this symbol,"},{"Start":"01:03.860 ","End":"01:10.445","Text":"is equal to n factorial over k factorial,"},{"Start":"01:10.445 ","End":"01:14.285","Text":"n minus k factorial."},{"Start":"01:14.285 ","End":"01:20.670","Text":"This together with this tells us how we expand a plus b to any power."},{"Start":"01:21.530 ","End":"01:24.715","Text":"Let\u0027s take n equals 4."},{"Start":"01:24.715 ","End":"01:28.625","Text":"We\u0027ve already done n goes from 0-3."},{"Start":"01:28.625 ","End":"01:32.360","Text":"If we go back, we did a plus b^3 and lower powers."},{"Start":"01:32.360 ","End":"01:36.080","Text":"Let\u0027s do a plus b^4 from the formula."},{"Start":"01:36.080 ","End":"01:44.065","Text":"First of all, I\u0027ll use the Sigma to say that it\u0027s Sigma where k goes from 0-4."},{"Start":"01:44.065 ","End":"01:47.895","Text":"Remember this means is going to be 5 terms,"},{"Start":"01:47.895 ","End":"01:49.560","Text":"0, 1, 2, 3,"},{"Start":"01:49.560 ","End":"01:51.180","Text":"4, always 1 more,"},{"Start":"01:51.180 ","End":"01:55.170","Text":"that\u0027s this off by 1 keeps recurring."},{"Start":"01:55.170 ","End":"01:58.335","Text":"N here is 4,"},{"Start":"01:58.335 ","End":"02:04.380","Text":"choose k, k is the index that varies from 0- 4,"},{"Start":"02:04.380 ","End":"02:09.575","Text":"a to the power of 4 minus k,"},{"Start":"02:09.575 ","End":"02:16.130","Text":"b to the power of k. The powers of b are the ones that increase from 0-4."},{"Start":"02:16.130 ","End":"02:20.585","Text":"The next step is to open up the Sigma,"},{"Start":"02:20.585 ","End":"02:22.444","Text":"so we have 5 terms."},{"Start":"02:22.444 ","End":"02:24.800","Text":"First one, when k is 0,"},{"Start":"02:24.800 ","End":"02:30.460","Text":"4 choose 0 a to the 4 minus 0 is 4,"},{"Start":"02:30.460 ","End":"02:33.710","Text":"and b to the 0, we don\u0027t write it."},{"Start":"02:33.710 ","End":"02:34.805","Text":"But we, I mean,"},{"Start":"02:34.805 ","End":"02:36.680","Text":"always this comes out to be 1,"},{"Start":"02:36.680 ","End":"02:39.445","Text":"but at this stage, we write it 4 choose 0."},{"Start":"02:39.445 ","End":"02:42.485","Text":"Next, 4 choose 1,"},{"Start":"02:42.485 ","End":"02:44.775","Text":"a to the 4 minus 1,"},{"Start":"02:44.775 ","End":"02:46.110","Text":"the a is decreasing,"},{"Start":"02:46.110 ","End":"02:47.160","Text":"the b is increasing."},{"Start":"02:47.160 ","End":"02:49.815","Text":"Here we have b to the 1 which is just b."},{"Start":"02:49.815 ","End":"02:52.630","Text":"Then, 4 choose 2."},{"Start":"02:52.630 ","End":"02:57.110","Text":"This time is squared and b increases squared."},{"Start":"02:57.110 ","End":"03:01.145","Text":"Then 4 choose 3, a^2,"},{"Start":"03:01.145 ","End":"03:02.810","Text":"so it\u0027s down to a,"},{"Start":"03:02.810 ","End":"03:06.625","Text":"b increases to the power of 3,"},{"Start":"03:06.625 ","End":"03:09.135","Text":"and then the last 1,"},{"Start":"03:09.135 ","End":"03:10.830","Text":"4 choose 4,"},{"Start":"03:10.830 ","End":"03:12.180","Text":"a to the power of 0,"},{"Start":"03:12.180 ","End":"03:14.040","Text":"so just don\u0027t write it,"},{"Start":"03:14.040 ","End":"03:19.825","Text":"and b^4, and of course the first and the last are 1,"},{"Start":"03:19.825 ","End":"03:22.580","Text":"that\u0027s always the case."},{"Start":"03:22.580 ","End":"03:27.055","Text":"Anyway, let\u0027s now compute these numbers."},{"Start":"03:27.055 ","End":"03:30.740","Text":"There is really two ways of going about this."},{"Start":"03:30.740 ","End":"03:34.115","Text":"One is just computing straight from the formula,"},{"Start":"03:34.115 ","End":"03:36.695","Text":"or we could use Pascal\u0027s triangle,"},{"Start":"03:36.695 ","End":"03:43.460","Text":"since we got the triangle up to the fourth row actually,"},{"Start":"03:43.460 ","End":"03:44.630","Text":"with the one with 3 in it,"},{"Start":"03:44.630 ","End":"03:45.830","Text":"we had a 1,"},{"Start":"03:45.830 ","End":"03:48.725","Text":"3, 3, 1,"},{"Start":"03:48.725 ","End":"03:50.735","Text":"above that we had 1,"},{"Start":"03:50.735 ","End":"03:52.340","Text":"2, 1,"},{"Start":"03:52.340 ","End":"03:54.425","Text":"and so on and so on."},{"Start":"03:54.425 ","End":"03:56.630","Text":"When we do in the next row,"},{"Start":"03:56.630 ","End":"04:00.125","Text":"we start with a 1 and we end with a 1."},{"Start":"04:00.125 ","End":"04:03.860","Text":"Remember each term now is the sum of the 2 above it,"},{"Start":"04:03.860 ","End":"04:06.260","Text":"so 1 and 3 is 4,"},{"Start":"04:06.260 ","End":"04:08.770","Text":"3 and 3 is 6,"},{"Start":"04:08.770 ","End":"04:11.905","Text":"3 and 1 is 4."},{"Start":"04:11.905 ","End":"04:14.465","Text":"The other way to do this,"},{"Start":"04:14.465 ","End":"04:19.684","Text":"to compute these numbers is to use this binomial,"},{"Start":"04:19.684 ","End":"04:26.330","Text":"this formula for the binomial coefficient and we\u0027ll do it this way also,"},{"Start":"04:26.330 ","End":"04:28.886","Text":"but it won\u0027t bother with the first and the last,"},{"Start":"04:28.886 ","End":"04:31.640","Text":"it works when k is 0 and when k is n,"},{"Start":"04:31.640 ","End":"04:35.070","Text":"but we\u0027ll just do these 3."},{"Start":"04:35.480 ","End":"04:41.130","Text":"4 choose 1 is 4 factorial over"},{"Start":"04:41.130 ","End":"04:50.280","Text":"1 factorial and 4 minus 1 is 3 factorial, that\u0027s here."},{"Start":"04:50.280 ","End":"04:52.890","Text":"I\u0027ll write them, and then we\u0027ll compute them,"},{"Start":"04:52.890 ","End":"05:00.780","Text":"4 choose 2 is 4 factorial over 2 factorial and then 4 minus 2,"},{"Start":"05:00.780 ","End":"05:04.305","Text":"which is also 2 factorial, and here,"},{"Start":"05:04.305 ","End":"05:08.685","Text":"4 factorial over 3 factorial,"},{"Start":"05:08.685 ","End":"05:13.660","Text":"4 minus 3 is 1 factorial."},{"Start":"05:14.390 ","End":"05:17.310","Text":"Now, 1 factorial is 1,"},{"Start":"05:17.310 ","End":"05:19.775","Text":"so it doesn\u0027t matter here and here."},{"Start":"05:19.775 ","End":"05:23.300","Text":"What we have, 4 factorial is 24,"},{"Start":"05:23.300 ","End":"05:25.235","Text":"3 factorial is 6,"},{"Start":"05:25.235 ","End":"05:29.245","Text":"24 over 6 is 4."},{"Start":"05:29.245 ","End":"05:34.380","Text":"This one is 24 over 2 times 2,"},{"Start":"05:34.380 ","End":"05:37.058","Text":"24 over 4 is 6,"},{"Start":"05:37.058 ","End":"05:40.535","Text":"and this is the same as this just different order,"},{"Start":"05:40.535 ","End":"05:42.245","Text":"so it\u0027s also 4,"},{"Start":"05:42.245 ","End":"05:44.930","Text":"and that\u0027s what we got here, here and here."},{"Start":"05:44.930 ","End":"05:48.710","Text":"Pascal\u0027s triangle method and"},{"Start":"05:48.710 ","End":"05:54.325","Text":"the formula method give us the same numbers and that\u0027s reassuring."},{"Start":"05:54.325 ","End":"05:56.925","Text":"I cleared some space here,"},{"Start":"05:56.925 ","End":"06:01.810","Text":"and now we can write the answer as a^4,"},{"Start":"06:01.810 ","End":"06:04.630","Text":"the 1 we don\u0027t write,"},{"Start":"06:04.630 ","End":"06:06.365","Text":"and then from here,"},{"Start":"06:06.365 ","End":"06:11.180","Text":"we have 4a^3b,"},{"Start":"06:11.180 ","End":"06:12.530","Text":"4 over 2,"},{"Start":"06:12.530 ","End":"06:21.540","Text":"4 choose 2 is 6a^2 b^2,"},{"Start":"06:21.550 ","End":"06:30.270","Text":"and then again 4 this time, ab^3,"},{"Start":"06:30.270 ","End":"06:35.565","Text":"and the last one 1 we don\u0027t write is just b^4,"},{"Start":"06:35.565 ","End":"06:44.050","Text":"so a plus b^4 is this."},{"Start":"06:44.270 ","End":"06:47.740","Text":"That\u0027s the example."},{"Start":"06:47.890 ","End":"06:52.010","Text":"Now let\u0027s look at some of the properties of"},{"Start":"06:52.010 ","End":"06:59.000","Text":"the binomial coefficient and this also relates to Pascal\u0027s triangle."},{"Start":"06:59.000 ","End":"07:03.050","Text":"I\u0027ll show you 3 properties,"},{"Start":"07:03.050 ","End":"07:11.260","Text":"the first property says n choose 0 is n choose n is 1."},{"Start":"07:11.260 ","End":"07:17.335","Text":"What this means is that the coefficients begin with 1 and end in 1."},{"Start":"07:17.335 ","End":"07:21.195","Text":"If n was like 4 here,"},{"Start":"07:21.195 ","End":"07:28.837","Text":"4 choose 0 is 1 and 4 choose 4 is also 1."},{"Start":"07:28.837 ","End":"07:33.698","Text":"What this is saying is that these are all 1\u0027s,"},{"Start":"07:33.698 ","End":"07:37.670","Text":"and these are all 1\u0027s."},{"Start":"07:37.670 ","End":"07:40.595","Text":"That\u0027s the first property."},{"Start":"07:40.595 ","End":"07:47.225","Text":"The next property I won\u0027t even read it out is what is written."},{"Start":"07:47.225 ","End":"07:50.119","Text":"Essentially, this is the symmetry property."},{"Start":"07:50.119 ","End":"07:56.120","Text":"Notice that if I put a vertical line through the middle, it\u0027s symmetrical."},{"Start":"07:56.120 ","End":"08:00.230","Text":"In our case, for example,"},{"Start":"08:00.230 ","End":"08:07.980","Text":"this came out to be 4 and this also came out to be 4."},{"Start":"08:08.150 ","End":"08:14.860","Text":"The middle one is its own mirror image if it\u0027s an odd number of times the middle one,"},{"Start":"08:14.860 ","End":"08:18.560","Text":"but you notice that if you go from left to right 1, 4,"},{"Start":"08:18.560 ","End":"08:20.570","Text":"6, 4, 1 or from right to left, 1,"},{"Start":"08:20.570 ","End":"08:22.580","Text":"4, 6, 4, 1, it\u0027s the same."},{"Start":"08:22.580 ","End":"08:24.335","Text":"This is what this says."},{"Start":"08:24.335 ","End":"08:25.910","Text":"In the particular example,"},{"Start":"08:25.910 ","End":"08:29.407","Text":"I chose n is 4, k is 1,"},{"Start":"08:29.407 ","End":"08:36.960","Text":"it says that 4 choose 1 is the same as 4 choose 4 minus 1, which is 3."},{"Start":"08:36.960 ","End":"08:42.740","Text":"In other words, if I subtract the 1 from the 4 and get 3 and put 3 here instead,"},{"Start":"08:42.740 ","End":"08:44.330","Text":"it\u0027s going to be the same."},{"Start":"08:44.330 ","End":"08:49.200","Text":"This is essentially what the symmetry is."},{"Start":"08:49.740 ","End":"08:57.040","Text":"The entries, the term is symmetrical if this plus this is 4 or n in general."},{"Start":"08:57.040 ","End":"09:00.070","Text":"Notice that 0 plus 4 is 4."},{"Start":"09:00.070 ","End":"09:02.050","Text":"1 plus 3 is 4."},{"Start":"09:02.050 ","End":"09:03.820","Text":"2 plus itself is 4."},{"Start":"09:03.820 ","End":"09:07.435","Text":"That\u0027s the symmetry condition and that\u0027s why it\u0027s n minus k,"},{"Start":"09:07.435 ","End":"09:09.535","Text":"together this and this adds up to n,"},{"Start":"09:09.535 ","End":"09:13.190","Text":"which means that they\u0027re symmetrically placed."},{"Start":"09:13.470 ","End":"09:16.630","Text":"As I said in the case of Pascal\u0027s triangle,"},{"Start":"09:16.630 ","End":"09:18.400","Text":"I mean you can look all over the place."},{"Start":"09:18.400 ","End":"09:19.720","Text":"5 is the same as 5,"},{"Start":"09:19.720 ","End":"09:21.220","Text":"10 is the same as 10,"},{"Start":"09:21.220 ","End":"09:23.500","Text":"28 here, 28 here,"},{"Start":"09:23.500 ","End":"09:25.540","Text":"and so on and so on."},{"Start":"09:25.540 ","End":"09:30.460","Text":"The 3rd property, this 1,"},{"Start":"09:30.460 ","End":"09:32.230","Text":"won\u0027t even read it out just yet,"},{"Start":"09:32.230 ","End":"09:37.945","Text":"but this is the mathematical form of that property,"},{"Start":"09:37.945 ","End":"09:39.670","Text":"let\u0027s take an example."},{"Start":"09:39.670 ","End":"09:43.990","Text":"Let\u0027s say I put a triangle here."},{"Start":"09:43.990 ","End":"09:48.955","Text":"Notice that we said that each entry is the sum of the 2 above it."},{"Start":"09:48.955 ","End":"09:51.505","Text":"That this is equal to this plus this."},{"Start":"09:51.505 ","End":"09:57.340","Text":"Well, what are these? This 21 is in row 7."},{"Start":"09:57.340 ","End":"09:59.680","Text":"It\u0027s the 8th row if you count this row,"},{"Start":"09:59.680 ","End":"10:02.860","Text":"but when I say row 7, I mean where the 7 is here."},{"Start":"10:02.860 ","End":"10:05.110","Text":"What this says is 21,"},{"Start":"10:05.110 ","End":"10:07.645","Text":"which is 7,"},{"Start":"10:07.645 ","End":"10:14.275","Text":"and the number below is 1 more,"},{"Start":"10:14.275 ","End":"10:16.765","Text":"or rather 1 less."},{"Start":"10:16.765 ","End":"10:20.125","Text":"You count 1, 2, 3,"},{"Start":"10:20.125 ","End":"10:23.935","Text":"the 3rd term, so the k is 2."},{"Start":"10:23.935 ","End":"10:25.870","Text":"Or you could start counting from 0."},{"Start":"10:25.870 ","End":"10:28.180","Text":"That\u0027s another way to do it, 0, 1, 2."},{"Start":"10:28.180 ","End":"10:31.850","Text":"That\u0027s 7, 2, that\u0027s the 21."},{"Start":"10:32.010 ","End":"10:38.450","Text":"The 35 is the following term is 7,"},{"Start":"10:38.610 ","End":"10:44.783","Text":"choose 3, because k is 0,"},{"Start":"10:44.783 ","End":"10:48.140","Text":"1, 2, 3, which is 1 more than the 2."},{"Start":"10:48.480 ","End":"10:57.550","Text":"The 56 is going to be the same index as the 35 because it\u0027s parallel."},{"Start":"10:57.550 ","End":"10:59.487","Text":"This is 0, 1, 2,"},{"Start":"10:59.487 ","End":"11:01.750","Text":"3, 0, 1, 2, 3."},{"Start":"11:01.750 ","End":"11:08.755","Text":"That would be 8 choose 3 because we\u0027re on the row below,"},{"Start":"11:08.755 ","End":"11:12.895","Text":"you can tell from the 8 here just like from the 7 here."},{"Start":"11:12.895 ","End":"11:14.365","Text":"It\u0027s 8 from here,"},{"Start":"11:14.365 ","End":"11:16.240","Text":"and then 0, 1, 2, 3,"},{"Start":"11:16.240 ","End":"11:18.010","Text":"or you count 1, 2, 3, 4,"},{"Start":"11:18.010 ","End":"11:20.275","Text":"but remember to subtract 1, so you get 3."},{"Start":"11:20.275 ","End":"11:23.350","Text":"That\u0027s 21, that\u0027s 35,"},{"Start":"11:23.350 ","End":"11:27.130","Text":"and this is 56."},{"Start":"11:27.130 ","End":"11:29.335","Text":"Just for extra practice,"},{"Start":"11:29.335 ","End":"11:33.475","Text":"let\u0027s compute them, 7 choose 2."},{"Start":"11:33.475 ","End":"11:40.660","Text":"This would be 7 factorial over 2 factorial."},{"Start":"11:40.660 ","End":"11:42.655","Text":"7 minus 2,"},{"Start":"11:42.655 ","End":"11:45.950","Text":"which is 5 factorial."},{"Start":"11:50.040 ","End":"11:53.380","Text":"You can compute it with the calculator,"},{"Start":"11:53.380 ","End":"11:56.260","Text":"but in these cases, it\u0027s best to just multiply out."},{"Start":"11:56.260 ","End":"12:03.130","Text":"7 factorial is 7 times 6 times 5 times 4 times 3 times 2 times 1."},{"Start":"12:03.130 ","End":"12:07.930","Text":"Then 2 factorial is 2 times 1,"},{"Start":"12:07.930 ","End":"12:13.390","Text":"and 5 factorial is 5 times 4 times 3 times 2 times 1."},{"Start":"12:13.390 ","End":"12:17.110","Text":"Notice that the 5 factorial and the 5 factorial cancel,"},{"Start":"12:17.110 ","End":"12:19.960","Text":"so all we\u0027re left with is this bit here,"},{"Start":"12:19.960 ","End":"12:22.255","Text":"7 times 6 over 2,"},{"Start":"12:22.255 ","End":"12:24.325","Text":"7 times 3,"},{"Start":"12:24.325 ","End":"12:27.685","Text":"that comes out to be 21."},{"Start":"12:27.685 ","End":"12:33.460","Text":"I\u0027ll leave you to check the formula that here we get 35,"},{"Start":"12:33.460 ","End":"12:37.100","Text":"and then here we get 56."},{"Start":"12:37.470 ","End":"12:40.300","Text":"I don\u0027t want to waste time doing that."},{"Start":"12:40.300 ","End":"12:43.105","Text":"It\u0027s just straightforward computation."},{"Start":"12:43.105 ","End":"12:47.470","Text":"Next is one more skill we need,"},{"Start":"12:47.470 ","End":"12:54.295","Text":"and that is how to find a particular term of a binomial expansion."},{"Start":"12:54.295 ","End":"12:59.425","Text":"Just like before, we have to figure out a plus b^4."},{"Start":"12:59.425 ","End":"13:01.495","Text":"Sometimes we just want,"},{"Start":"13:01.495 ","End":"13:04.700","Text":"I don\u0027t know the 3rd term."},{"Start":"13:07.500 ","End":"13:11.755","Text":"I want to remind you again of the off by 1 thing."},{"Start":"13:11.755 ","End":"13:18.235","Text":"Meaning that this binomial coefficient,"},{"Start":"13:18.235 ","End":"13:22.975","Text":"n choose k a^n minus k b^k is not the kth term,"},{"Start":"13:22.975 ","End":"13:25.510","Text":"it\u0027s the k plus first term."},{"Start":"13:25.510 ","End":"13:28.190","Text":"Does this off by 1 thing."},{"Start":"13:30.870 ","End":"13:33.835","Text":"The example is,"},{"Start":"13:33.835 ","End":"13:37.435","Text":"find the 6th term in the expansion,"},{"Start":"13:37.435 ","End":"13:40.135","Text":"and here I made it slightly different."},{"Start":"13:40.135 ","End":"13:41.740","Text":"Instead of a plus b,"},{"Start":"13:41.740 ","End":"13:43.975","Text":"I got fed up of a plus b,"},{"Start":"13:43.975 ","End":"13:46.780","Text":"to show you that it doesn\u0027t have to be those 2 letters."},{"Start":"13:46.780 ","End":"13:49.710","Text":"Let\u0027s do x plus y^15."},{"Start":"13:49.710 ","End":"13:52.320","Text":"We want the 6th term."},{"Start":"13:52.320 ","End":"13:56.030","Text":"Now, 6 is 5 plus 1."},{"Start":"13:56.030 ","End":"14:00.910","Text":"What we\u0027re talking about is k equals 5."},{"Start":"14:00.910 ","End":"14:02.941","Text":"6 meaning 0,"},{"Start":"14:02.941 ","End":"14:04.675","Text":"1, 2, 3, 4, 5."},{"Start":"14:04.675 ","End":"14:07.255","Text":"We count 6 starting from 0."},{"Start":"14:07.255 ","End":"14:13.870","Text":"The n, in this case, is 15."},{"Start":"14:13.870 ","End":"14:23.470","Text":"What we want, the term will be 15 choose 5,"},{"Start":"14:23.470 ","End":"14:27.940","Text":"that\u0027s the n choose k. Then we\u0027ll want,"},{"Start":"14:27.940 ","End":"14:31.990","Text":"not a, but we\u0027ll have x instead."},{"Start":"14:31.990 ","End":"14:35.530","Text":"Of course, if you have different letters, it\u0027s different here."},{"Start":"14:35.530 ","End":"14:45.415","Text":"It\u0027s x^n minus k. 15 minus 5 is 10 and here,"},{"Start":"14:45.415 ","End":"14:50.260","Text":"b which is y^k, is this."},{"Start":"14:50.260 ","End":"14:52.120","Text":"This is what we want."},{"Start":"14:52.120 ","End":"14:58.375","Text":"The only thing missing is the actual computation of 15 choose 5."},{"Start":"14:58.375 ","End":"15:05.695","Text":"That would be 15 factorial over 5 factorial."},{"Start":"15:05.695 ","End":"15:10.495","Text":"Then 15 minus 5 is 10 factorial."},{"Start":"15:10.495 ","End":"15:12.520","Text":"As we saw before,"},{"Start":"15:12.520 ","End":"15:15.955","Text":"a whole chunk will cancel if we write out the factors."},{"Start":"15:15.955 ","End":"15:23.380","Text":"We can write 15 times 14 times 13 times 12 times 11 times."},{"Start":"15:23.380 ","End":"15:26.795","Text":"Notice what comes after the 11, we get 10, 9,"},{"Start":"15:26.795 ","End":"15:30.865","Text":"8, and so on,"},{"Start":"15:30.865 ","End":"15:33.400","Text":"right down to 1."},{"Start":"15:33.400 ","End":"15:37.960","Text":"On the denominator, we get 5 factorial,"},{"Start":"15:37.960 ","End":"15:43.225","Text":"which is 5 times 4 times 3 times 2 times 1."},{"Start":"15:43.225 ","End":"15:45.490","Text":"After that, we get the 10 factorial,"},{"Start":"15:45.490 ","End":"15:47.305","Text":"which is 10, 9, 8,"},{"Start":"15:47.305 ","End":"15:49.690","Text":"7, 6, down to 1."},{"Start":"15:49.690 ","End":"15:52.450","Text":"This part cancels,"},{"Start":"15:52.450 ","End":"15:55.015","Text":"and we\u0027re left with just this."},{"Start":"15:55.015 ","End":"15:57.730","Text":"Of course, we could do this on the calculator,"},{"Start":"15:57.730 ","End":"16:01.660","Text":"but let\u0027s see if we can cancel a bit like, be creative."},{"Start":"16:01.660 ","End":"16:05.575","Text":"15 is 5 times 3."},{"Start":"16:05.575 ","End":"16:13.135","Text":"What else? 4 will go into 12,"},{"Start":"16:13.135 ","End":"16:21.235","Text":"3 times and 2 will go into 14, 7 times."},{"Start":"16:21.235 ","End":"16:24.625","Text":"What are we left with all the denominators gone,"},{"Start":"16:24.625 ","End":"16:29.620","Text":"and we\u0027re left with 7 times 13 times"},{"Start":"16:29.620 ","End":"16:35.590","Text":"3 times 11, and I don\u0027t know."},{"Start":"16:35.590 ","End":"16:39.550","Text":"7 times 13 is 91,"},{"Start":"16:39.550 ","End":"16:43.240","Text":"3 times 11 is 33."},{"Start":"16:43.240 ","End":"16:46.105","Text":"I\u0027m going to need a calculator."},{"Start":"16:46.105 ","End":"16:50.780","Text":"It comes out to be 3,003."},{"Start":"16:50.970 ","End":"16:54.100","Text":"All we have to do is plug that here,"},{"Start":"16:54.100 ","End":"16:58.400","Text":"and the answer is,"},{"Start":"16:58.500 ","End":"17:00.595","Text":"well, I\u0027ll write it over here,"},{"Start":"17:00.595 ","End":"17:10.690","Text":"3,003 times x^10 y^5."},{"Start":"17:10.690 ","End":"17:13.285","Text":"Highlight that."},{"Start":"17:13.285 ","End":"17:15.460","Text":"With this example,"},{"Start":"17:15.460 ","End":"17:18.670","Text":"that concludes the tutorial."},{"Start":"17:18.670 ","End":"17:24.865","Text":"There\u0027s a lot of solved exercises following."},{"Start":"17:24.865 ","End":"17:27.115","Text":"You should continue with those."},{"Start":"17:27.115 ","End":"17:29.480","Text":"Here we\u0027re done."}],"Thumbnail":null,"ID":26921},{"Watched":false,"Name":"Exercise 1","Duration":"5m 17s","ChapterTopicVideoID":13481,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this exercise, which is really four exercises in one,"},{"Start":"00:03.450 ","End":"00:06.030","Text":"we have to calculate the binomial coefficients."},{"Start":"00:06.030 ","End":"00:08.220","Text":"Let\u0027s remember the formula,"},{"Start":"00:08.220 ","End":"00:14.025","Text":"which is really a definition that if we have n here and k here,"},{"Start":"00:14.025 ","End":"00:24.615","Text":"then it\u0027s equal to n factorial over k factorial times n minus k factorial."},{"Start":"00:24.615 ","End":"00:29.325","Text":"Straightforward application of the formula."},{"Start":"00:29.325 ","End":"00:33.765","Text":"In a, we have 5, 3,"},{"Start":"00:33.765 ","End":"00:37.815","Text":"so it\u0027s 5 factorial over"},{"Start":"00:37.815 ","End":"00:44.205","Text":"3 factorial and usually mentally do the 5 minus 3,"},{"Start":"00:44.205 ","End":"00:46.310","Text":"which is 2 factorial."},{"Start":"00:46.310 ","End":"00:49.085","Text":"Why write another step?"},{"Start":"00:49.085 ","End":"00:51.960","Text":"This gives us 5 factorial,"},{"Start":"00:51.960 ","End":"00:56.745","Text":"is 5 times 4 times 3 times 2 times 1."},{"Start":"00:56.745 ","End":"00:59.240","Text":"Now it\u0027s silly to write the times 1,"},{"Start":"00:59.240 ","End":"01:04.715","Text":"but it\u0027s just the way we do it because it\u0027s like a pattern from 5 down to 1."},{"Start":"01:04.715 ","End":"01:09.358","Text":"Then 3 factorial, 3 times 2 times 1,"},{"Start":"01:09.358 ","End":"01:12.835","Text":"and 2 factorial, 2 times 1."},{"Start":"01:12.835 ","End":"01:15.795","Text":"Now, some things cancel,"},{"Start":"01:15.795 ","End":"01:18.360","Text":"2 and 1, 2 and 1,"},{"Start":"01:18.360 ","End":"01:21.255","Text":"this 3 with this 3,"},{"Start":"01:21.255 ","End":"01:27.225","Text":"and I guess 2 goes into 4 twice,"},{"Start":"01:27.225 ","End":"01:30.090","Text":"and so the answer is 10."},{"Start":"01:30.090 ","End":"01:35.640","Text":"Now part b, 8, 3."},{"Start":"01:35.640 ","End":"01:39.960","Text":"8 factorial, 3 factorial,"},{"Start":"01:39.960 ","End":"01:41.760","Text":"and the 8 minus 3,"},{"Start":"01:41.760 ","End":"01:46.220","Text":"do in our heads, is 5 factorial."},{"Start":"01:46.220 ","End":"01:48.170","Text":"If we were really spelling it out,"},{"Start":"01:48.170 ","End":"01:51.215","Text":"we\u0027d have an extra step where instead of 5 factorial,"},{"Start":"01:51.215 ","End":"01:54.875","Text":"we\u0027d have 8 minus 3 factorial."},{"Start":"01:54.875 ","End":"01:57.365","Text":"But as I said, why waste the step,"},{"Start":"01:57.365 ","End":"02:01.205","Text":"because the subtraction part is the easy part."},{"Start":"02:01.205 ","End":"02:06.990","Text":"So 8 times 7 times 6 times 5 times 4 times"},{"Start":"02:06.990 ","End":"02:13.045","Text":"3 times 2 times 1 over 3 times 2 times 1,"},{"Start":"02:13.045 ","End":"02:14.915","Text":"5, 4,"},{"Start":"02:14.915 ","End":"02:17.850","Text":"3, 2, 1."},{"Start":"02:17.850 ","End":"02:21.150","Text":"This whole bit cancels with this."},{"Start":"02:21.150 ","End":"02:25.260","Text":"Look, 3 times 2 is 6."},{"Start":"02:25.260 ","End":"02:31.945","Text":"What we\u0027re left with is 8 times 7, which is 56."},{"Start":"02:31.945 ","End":"02:36.260","Text":"Now the next one, 10 over 1."},{"Start":"02:36.260 ","End":"02:42.590","Text":"I\u0027m used to saying 10 choose 1 because these are also from combinatorics."},{"Start":"02:42.590 ","End":"02:44.615","Text":"Anyway, it doesn\u0027t matter how you pronounce it."},{"Start":"02:44.615 ","End":"02:46.550","Text":"That\u0027s how it\u0027s written."},{"Start":"02:46.550 ","End":"02:52.655","Text":"That is equal to 10 factorial over 1 factorial,"},{"Start":"02:52.655 ","End":"02:54.080","Text":"10 minus 1,"},{"Start":"02:54.080 ","End":"02:57.495","Text":"do it mentally, 9 factorial."},{"Start":"02:57.495 ","End":"02:58.995","Text":"We get 10,"},{"Start":"02:58.995 ","End":"03:00.885","Text":"9, 8,"},{"Start":"03:00.885 ","End":"03:02.130","Text":"7, 6,"},{"Start":"03:02.130 ","End":"03:03.360","Text":"5, 4,"},{"Start":"03:03.360 ","End":"03:04.770","Text":"3, 2,"},{"Start":"03:04.770 ","End":"03:12.210","Text":"1 over 1 factorial is just 1 and then 9,"},{"Start":"03:12.210 ","End":"03:14.505","Text":"8, 7, 6,"},{"Start":"03:14.505 ","End":"03:16.110","Text":"5, 4,"},{"Start":"03:16.110 ","End":"03:18.570","Text":"3, 2, 1."},{"Start":"03:18.570 ","End":"03:21.900","Text":"This whole bit cancels with this,"},{"Start":"03:21.900 ","End":"03:24.465","Text":"and what we\u0027re left with is 10 over 1,"},{"Start":"03:24.465 ","End":"03:26.695","Text":"which is simply 10."},{"Start":"03:26.695 ","End":"03:28.250","Text":"Now the last part,"},{"Start":"03:28.250 ","End":"03:32.480","Text":"which is the only part which is abstract or general,"},{"Start":"03:32.480 ","End":"03:34.310","Text":"not specific numbers,"},{"Start":"03:34.310 ","End":"03:37.195","Text":"n over n minus 1."},{"Start":"03:37.195 ","End":"03:39.720","Text":"That\u0027s part d,"},{"Start":"03:39.720 ","End":"03:42.140","Text":"n here, n minus 1 here."},{"Start":"03:42.140 ","End":"03:44.195","Text":"Now we\u0027ve lost the formula,"},{"Start":"03:44.195 ","End":"03:45.620","Text":"but it\u0027s back there."},{"Start":"03:45.620 ","End":"03:47.870","Text":"We take the factorial of this."},{"Start":"03:47.870 ","End":"03:49.505","Text":"You should memorize that formula."},{"Start":"03:49.505 ","End":"03:51.082","Text":"I mean, that\u0027s the definition,"},{"Start":"03:51.082 ","End":"03:53.420","Text":"you can\u0027t go very far without that formula,"},{"Start":"03:53.420 ","End":"03:57.425","Text":"over this factorial, which is n minus 1 factorial."},{"Start":"03:57.425 ","End":"04:02.195","Text":"This time, I will write out the subtraction before I actually do the subtraction."},{"Start":"04:02.195 ","End":"04:05.635","Text":"So it\u0027s n, the top minus the bottom,"},{"Start":"04:05.635 ","End":"04:07.020","Text":"n minus 1,"},{"Start":"04:07.020 ","End":"04:10.110","Text":"and I need a pair of brackets here and factorial."},{"Start":"04:10.110 ","End":"04:16.145","Text":"That\u0027s equal to n minus n minus 1 is just 1 factorial."},{"Start":"04:16.145 ","End":"04:18.020","Text":"We have n factorial,"},{"Start":"04:18.020 ","End":"04:21.020","Text":"which is n, n minus 1."},{"Start":"04:21.020 ","End":"04:22.490","Text":"We have to put dot, dot,"},{"Start":"04:22.490 ","End":"04:24.665","Text":"dot because we don\u0027t know what n is."},{"Start":"04:24.665 ","End":"04:27.790","Text":"Down to 2 times 1."},{"Start":"04:27.790 ","End":"04:33.050","Text":"Then n minus 1 factorial is just the same thing,"},{"Start":"04:33.050 ","End":"04:36.710","Text":"but beginning with n minus 1 and minus 2,"},{"Start":"04:36.710 ","End":"04:40.230","Text":"and so on down to 2 times 1."},{"Start":"04:40.230 ","End":"04:43.060","Text":"Then 1 factorial, which is just 1."},{"Start":"04:43.060 ","End":"04:47.138","Text":"Now this whole bit cancels with this whole bit,"},{"Start":"04:47.138 ","End":"04:50.850","Text":"and what we\u0027re left with is n over 1,"},{"Start":"04:50.850 ","End":"04:56.152","Text":"which is just n. Just to give you a concrete example,"},{"Start":"04:56.152 ","End":"05:02.840","Text":"suppose I had 15 over 14,"},{"Start":"05:02.840 ","End":"05:05.220","Text":"this is what pattern means,"},{"Start":"05:05.220 ","End":"05:07.670","Text":"what\u0027s below is 1 less than what\u0027s above,"},{"Start":"05:07.670 ","End":"05:11.780","Text":"then we know that the answer is the one on the top that is equal to 15."},{"Start":"05:11.780 ","End":"05:13.280","Text":"That\u0027s just an example."},{"Start":"05:13.280 ","End":"05:17.460","Text":"Okay, we\u0027re done with this exercise."}],"Thumbnail":null,"ID":14122},{"Watched":false,"Name":"Exercise 2","Duration":"1m 29s","ChapterTopicVideoID":13482,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:04.200","Text":"In this exercise, we have another proof question"},{"Start":"00:04.200 ","End":"00:07.530","Text":"but it\u0027s not really as tough as a proof question,"},{"Start":"00:07.530 ","End":"00:10.425","Text":"just a matter of substituting in the formula."},{"Start":"00:10.425 ","End":"00:12.330","Text":"Perhaps a right reminder,"},{"Start":"00:12.330 ","End":"00:14.905","Text":"if we have n over k,"},{"Start":"00:14.905 ","End":"00:16.520","Text":"I\u0027m not really sure how to pronounce this,"},{"Start":"00:16.520 ","End":"00:19.040","Text":"I say n choose k from combinatorics."},{"Start":"00:19.040 ","End":"00:23.405","Text":"Anyway, it\u0027s equal by definition to n factorial"},{"Start":"00:23.405 ","End":"00:29.310","Text":"over k factorial times n minus k factorial,"},{"Start":"00:29.310 ","End":"00:31.095","Text":"so applying that here,"},{"Start":"00:31.095 ","End":"00:33.405","Text":"we would get n,"},{"Start":"00:33.405 ","End":"00:37.725","Text":"n is equal to n factorial over."},{"Start":"00:37.725 ","End":"00:46.535","Text":"Now n is k, so that\u0027s n factorial and then n minus k factorial is n minus n factorial."},{"Start":"00:46.535 ","End":"00:54.410","Text":"Now n factorial cancels with n factorial and 0 factorial is just 1,"},{"Start":"00:54.410 ","End":"00:57.275","Text":"and so we end up with 1."},{"Start":"00:57.275 ","End":"01:01.370","Text":"This basically means that when we take a plus b to the power of n,"},{"Start":"01:01.370 ","End":"01:04.460","Text":"the first term is a to the n and its coefficient is 1,"},{"Start":"01:04.460 ","End":"01:07.190","Text":"and the left term is b to the n. This coefficient is 1,"},{"Start":"01:07.190 ","End":"01:09.515","Text":"the first and last coefficients are 1."},{"Start":"01:09.515 ","End":"01:12.925","Text":"Let\u0027s do the other one, n, 0,"},{"Start":"01:12.925 ","End":"01:21.365","Text":"so n factorial over 0 factorial n minus 0 factorial."},{"Start":"01:21.365 ","End":"01:23.030","Text":"Well, so same thing as what we had above,"},{"Start":"01:23.030 ","End":"01:24.560","Text":"0 factorial here and here,"},{"Start":"01:24.560 ","End":"01:26.270","Text":"n factorial here and here."},{"Start":"01:26.270 ","End":"01:30.060","Text":"This will also equal 1, and that\u0027s it."}],"Thumbnail":null,"ID":14123},{"Watched":false,"Name":"Exercise 3","Duration":"6m 24s","ChapterTopicVideoID":13483,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"This exercise is a proof question,"},{"Start":"00:03.210 ","End":"00:09.270","Text":"and it turns out to be very useful for actually computing these binomial coefficients."},{"Start":"00:09.270 ","End":"00:13.590","Text":"What we have to show is that for all n and for all k,"},{"Start":"00:13.590 ","End":"00:15.135","Text":"which is less than n,"},{"Start":"00:15.135 ","End":"00:17.940","Text":"though typically we won\u0027t take 0 and n,"},{"Start":"00:17.940 ","End":"00:21.330","Text":"we\u0027ll take it between 1 and n-1."},{"Start":"00:21.330 ","End":"00:24.660","Text":"But theoretically, will work for the endpoints also."},{"Start":"00:24.660 ","End":"00:30.005","Text":"We have to show that this binomial coefficient can be described in two different ways."},{"Start":"00:30.005 ","End":"00:33.410","Text":"One is using (k) factorial in the denominator,"},{"Start":"00:33.410 ","End":"00:35.885","Text":"the other uses (n-k) factorial,"},{"Start":"00:35.885 ","End":"00:40.410","Text":"and in each case, we start from n and work our way down."},{"Start":"00:40.490 ","End":"00:43.670","Text":"An example will illustrate this best."},{"Start":"00:43.670 ","End":"00:46.055","Text":"What I\u0027m going to say is that the sustained number"},{"Start":"00:46.055 ","End":"00:48.725","Text":"of factors on the top and the bottom. But we\u0027ll see this."},{"Start":"00:48.725 ","End":"00:53.160","Text":"I\u0027ll take the example of 8 choose 3."},{"Start":"00:53.160 ","End":"00:55.985","Text":"Now, if I use the first formula,"},{"Start":"00:55.985 ","End":"01:00.295","Text":"it says 8.7,"},{"Start":"01:00.295 ","End":"01:02.840","Text":"and what we have to do is go down,"},{"Start":"01:02.840 ","End":"01:06.860","Text":"up to n-k on the bottom will be 5,"},{"Start":"01:06.860 ","End":"01:09.920","Text":"and we\u0027ll go down to k+1, which is 4."},{"Start":"01:09.920 ","End":"01:14.375","Text":"So, it\u0027s 8.7.6.5.4,"},{"Start":"01:14.375 ","End":"01:19.530","Text":"on the bottom, it\u0027s (5)factorial 5.4.3.2.1."},{"Start":"01:20.140 ","End":"01:24.725","Text":"Notice that they really are lined up in pairs, the factors."},{"Start":"01:24.725 ","End":"01:28.190","Text":"This will help us to figure out the numerator."},{"Start":"01:28.190 ","End":"01:33.320","Text":"We won\u0027t have to figure out what is k+1 and all that will just write (5)factorial."},{"Start":"01:33.320 ","End":"01:37.880","Text":"The other expression is k factorial,"},{"Start":"01:37.880 ","End":"01:41.615","Text":"which is (3)factorial, which is 3.2.1."},{"Start":"01:41.615 ","End":"01:44.060","Text":"On the top, like I said,"},{"Start":"01:44.060 ","End":"01:45.380","Text":"you don\u0027t have to interpret this."},{"Start":"01:45.380 ","End":"01:51.410","Text":"You could just write starting from a to just keep going down until you hit the last one."},{"Start":"01:51.410 ","End":"01:56.600","Text":"But if you use the formula, this is n-k+1."},{"Start":"01:56.600 ","End":"01:58.490","Text":"8-3+1 is indeed 6."},{"Start":"01:58.490 ","End":"02:00.710","Text":"Now, why is this so?"},{"Start":"02:00.710 ","End":"02:03.035","Text":"Let\u0027s look at it in the example."},{"Start":"02:03.035 ","End":"02:04.715","Text":"If we\u0027re using the formula,"},{"Start":"02:04.715 ","End":"02:06.380","Text":"then we would say it\u0027s"},{"Start":"02:06.380 ","End":"02:16.520","Text":"8.7.6.5.4.3.2.1 over (3)factorial,"},{"Start":"02:16.520 ","End":"02:20.500","Text":"(5)factorial, 3.2.1, I would like to just brackets for emphasis,"},{"Start":"02:20.500 ","End":"02:25.070","Text":"and 5 factorial, 5.4.3.2.1."},{"Start":"02:25.070 ","End":"02:30.935","Text":"Now, I can cancel either one of these strings of factors."},{"Start":"02:30.935 ","End":"02:35.280","Text":"If we cancel the first string, 3.2.1,"},{"Start":"02:35.280 ","End":"02:41.570","Text":"with 3.2.1, then we get the first expression."},{"Start":"02:41.570 ","End":"02:46.230","Text":"Notice it\u0027s 8 down to 4 and then 5 factorial."},{"Start":"02:46.230 ","End":"02:49.490","Text":"But if instead of doing this,"},{"Start":"02:49.490 ","End":"02:54.500","Text":"we cancel this and then we go all the way up to 5."},{"Start":"02:54.500 ","End":"02:57.230","Text":"Then we would get this expression."},{"Start":"02:57.230 ","End":"03:01.750","Text":"We get 8.7.6 over 3.2.1."},{"Start":"03:01.750 ","End":"03:07.610","Text":"Now, typically we choose the one that is smaller, has less factors."},{"Start":"03:07.610 ","End":"03:11.180","Text":"So I would say which is smaller k or n-k."},{"Start":"03:11.180 ","End":"03:13.310","Text":"In this case, 3 is less than 8."},{"Start":"03:13.310 ","End":"03:15.800","Text":"So we would go with this one,"},{"Start":"03:15.800 ","End":"03:17.015","Text":"which is this one."},{"Start":"03:17.015 ","End":"03:18.275","Text":"But if it was say,"},{"Start":"03:18.275 ","End":"03:22.625","Text":"5 here, then we\u0027d go with this one."},{"Start":"03:22.625 ","End":"03:26.735","Text":"Now let\u0027s do the general case and we\u0027ll prove it."},{"Start":"03:26.735 ","End":"03:30.680","Text":"Let\u0027s start off with the definition,"},{"Start":"03:30.680 ","End":"03:37.165","Text":"n over k or n choose k is n(n-1),"},{"Start":"03:37.165 ","End":"03:44.730","Text":"and so on, down to 1 or 2 times 1 over."},{"Start":"03:44.730 ","End":"03:48.660","Text":"Then we want (k) factorial,"},{"Start":"03:48.660 ","End":"03:52.035","Text":"which is k,"},{"Start":"03:52.035 ","End":"03:58.110","Text":"and so on down to 1 and that\u0027s like here."},{"Start":"03:58.110 ","End":"04:01.530","Text":"Then the other one is (n-k) factorial."},{"Start":"04:01.530 ","End":"04:04.890","Text":"So it\u0027s n-k,"},{"Start":"04:04.890 ","End":"04:07.285","Text":"each time decreasing by 1."},{"Start":"04:07.285 ","End":"04:08.780","Text":"I won\u0027t write the middle bits."},{"Start":"04:08.780 ","End":"04:11.390","Text":"It\u0027s just from n-k down to 1."},{"Start":"04:11.390 ","End":"04:13.325","Text":"On the denominator,"},{"Start":"04:13.325 ","End":"04:16.530","Text":"we\u0027ll just leave it as in the formula,"},{"Start":"04:17.260 ","End":"04:20.580","Text":"(k) factorial, n-k factorial."},{"Start":"04:20.580 ","End":"04:27.245","Text":"Now what I\u0027m going to do is rewrite it just like we did here in two different ways."},{"Start":"04:27.245 ","End":"04:31.740","Text":"The first way I want the (k) factorial to cancel."},{"Start":"04:31.940 ","End":"04:34.770","Text":"We\u0027ll just split it up into two."},{"Start":"04:34.770 ","End":"04:41.760","Text":"On the one hand, it equals n(n-1) and I want the (k) factorial to cancel."},{"Start":"04:41.760 ","End":"04:46.650","Text":"So I\u0027ll stop one short of k. So we\u0027ll"},{"Start":"04:46.650 ","End":"04:52.455","Text":"get k(k-1) down to 1."},{"Start":"04:52.455 ","End":"04:57.510","Text":"The one before k would be k+1."},{"Start":"04:57.510 ","End":"05:03.160","Text":"So we can see that this part is (k) factorial."},{"Start":"05:03.160 ","End":"05:05.195","Text":"Now on the denominator,"},{"Start":"05:05.195 ","End":"05:12.865","Text":"we had (k) factorial and (n-k) factorial."},{"Start":"05:12.865 ","End":"05:18.150","Text":"Then this cancels with this and what we\u0027re left"},{"Start":"05:18.150 ","End":"05:23.735","Text":"with is the first formula, this one here."},{"Start":"05:23.735 ","End":"05:25.790","Text":"Compare it, it\u0027s the same."},{"Start":"05:25.790 ","End":"05:29.060","Text":"On the other hand, we could split this numerator differently."},{"Start":"05:29.060 ","End":"05:37.294","Text":"We could go n(n-1) and this time I want the (n-k) factorial to cancel."},{"Start":"05:37.294 ","End":"05:38.990","Text":"So let\u0027s see where we will continue."},{"Start":"05:38.990 ","End":"05:44.240","Text":"I need to have (n-k) factorial,"},{"Start":"05:44.240 ","End":"05:51.840","Text":"(n-k)(n-k-1), and so on down to 1."},{"Start":"05:51.840 ","End":"06:01.330","Text":"So the one before that would be n-k+1 and on the denominator,"},{"Start":"06:02.030 ","End":"06:05.430","Text":"(k) factorial, (n-k) factorial."},{"Start":"06:05.430 ","End":"06:08.210","Text":"This time we\u0027ll cancel differently."},{"Start":"06:08.210 ","End":"06:14.580","Text":"This bit will go with this and what we\u0027re left with n down"},{"Start":"06:14.580 ","End":"06:20.870","Text":"to n-k+1 over (k) factorial is exactly this bit."},{"Start":"06:20.870 ","End":"06:23.630","Text":"So that proves the second half,"},{"Start":"06:23.630 ","End":"06:25.740","Text":"and that\u0027s it. We\u0027re done."}],"Thumbnail":null,"ID":14124},{"Watched":false,"Name":"Exercise 4","Duration":"5m 58s","ChapterTopicVideoID":13484,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.320","Text":"In this exercise, we have to use Newton\u0027s binomial expansion to write out (x + y)^6,"},{"Start":"00:07.320 ","End":"00:09.075","Text":"I\u0027ll just write out the formula,"},{"Start":"00:09.075 ","End":"00:10.440","Text":"at least in one version,"},{"Start":"00:10.440 ","End":"00:14.775","Text":"it\u0027s written as (a + b)^n,"},{"Start":"00:14.775 ","End":"00:19.680","Text":"is the sum k goes from 0-n,"},{"Start":"00:19.680 ","End":"00:21.870","Text":"n choose k,"},{"Start":"00:21.870 ","End":"00:28.155","Text":"a^n minus k, b^k."},{"Start":"00:28.155 ","End":"00:30.240","Text":"Replace a and b by x and y,"},{"Start":"00:30.240 ","End":"00:32.595","Text":"and replace n by 6."},{"Start":"00:32.595 ","End":"00:42.360","Text":"What we\u0027ll get is that this equal to the sum k goes from 0-6."},{"Start":"00:42.360 ","End":"00:46.690","Text":"6, choose k,"},{"Start":"00:48.140 ","End":"00:55.710","Text":"x^6 minus k. That\u0027s this bit, y^k."},{"Start":"00:55.710 ","End":"00:58.620","Text":"Now let\u0027s see. I want to drop this sigma,"},{"Start":"00:58.620 ","End":"01:01.980","Text":"and we\u0027ll get actually 7 terms from 0-6."},{"Start":"01:01.980 ","End":"01:10.560","Text":"If k is 0, we get 6,0, x^6, y^0."},{"Start":"01:10.560 ","End":"01:17.035","Text":"Next, 6,1, x^5, y^1."},{"Start":"01:17.035 ","End":"01:21.230","Text":"Each time, reducing something by 1, or increasing by 1."},{"Start":"01:21.230 ","End":"01:25.595","Text":"Here, 6,2, x^4, y^2,"},{"Start":"01:25.595 ","End":"01:28.775","Text":"plus then we need 6,3,"},{"Start":"01:28.775 ","End":"01:30.830","Text":"x^3, y^3,"},{"Start":"01:30.830 ","End":"01:35.670","Text":"6,4, x^2, y^4."},{"Start":"01:35.670 ","End":"01:38.590","Text":"The other powers of y are increasing."},{"Start":"01:38.660 ","End":"01:45.405","Text":"Then 6,5, x^1, y^5,"},{"Start":"01:45.405 ","End":"01:53.680","Text":"and finally 6,6, x^0, y^6."},{"Start":"01:53.680 ","End":"01:55.760","Text":"Now to calculate these coefficients,"},{"Start":"01:55.760 ","End":"01:58.685","Text":"I would like to use one of the previous problems,"},{"Start":"01:58.685 ","End":"02:03.145","Text":"which gave an alternate form for n choose k,"},{"Start":"02:03.145 ","End":"02:05.820","Text":"as on the denominator,"},{"Start":"02:05.820 ","End":"02:08.670","Text":"n - k factorial,"},{"Start":"02:08.670 ","End":"02:10.110","Text":"and on the numerator,"},{"Start":"02:10.110 ","End":"02:13.710","Text":"n times n minus 1 down,"},{"Start":"02:13.710 ","End":"02:16.410","Text":"up to dot,"},{"Start":"02:16.410 ","End":"02:20.695","Text":"the last one would be k plus 1."},{"Start":"02:20.695 ","End":"02:23.795","Text":"It turns out that the same number of terms here and here."},{"Start":"02:23.795 ","End":"02:26.390","Text":"Let\u0027s use that. Now the first,"},{"Start":"02:26.390 ","End":"02:27.920","Text":"or the last coefficient, is always 1,"},{"Start":"02:27.920 ","End":"02:29.885","Text":"so there\u0027s no point wasting time with that."},{"Start":"02:29.885 ","End":"02:34.320","Text":"This is 1, this is x^6, y^0 is 1."},{"Start":"02:34.320 ","End":"02:35.790","Text":"Now the next one,"},{"Start":"02:35.790 ","End":"02:37.935","Text":"6 choose 1,"},{"Start":"02:37.935 ","End":"02:43.195","Text":"means that we get using this formula on the bottom, we need."},{"Start":"02:43.195 ","End":"02:46.400","Text":"Yes, you should have written the other form of this."},{"Start":"02:46.400 ","End":"02:49.430","Text":"The other form is, we\u0027re on the bottom,"},{"Start":"02:49.430 ","End":"02:52.730","Text":"we put k factorial instead of n minus k factorial,"},{"Start":"02:52.730 ","End":"02:55.655","Text":"and here we have n minus 1,"},{"Start":"02:55.655 ","End":"02:58.680","Text":"and it goes down to, k plus 1."},{"Start":"02:58.680 ","End":"03:03.060","Text":"This time we have n minus k plus 1."},{"Start":"03:03.060 ","End":"03:08.540","Text":"The one I would suggest using is the one where the denominator is the smallest,"},{"Start":"03:08.540 ","End":"03:13.670","Text":"whatever smaller k or n minus k. Back here,"},{"Start":"03:13.670 ","End":"03:15.350","Text":"for the first and the last,"},{"Start":"03:15.350 ","End":"03:17.275","Text":"we always know it\u0027s going to be 1,"},{"Start":"03:17.275 ","End":"03:18.500","Text":"so this one and this one,"},{"Start":"03:18.500 ","End":"03:20.720","Text":"we don\u0027t use the formula for this."},{"Start":"03:20.720 ","End":"03:23.695","Text":"This is 1, this is x^6,"},{"Start":"03:23.695 ","End":"03:25.260","Text":"and y^0 is 1,"},{"Start":"03:25.260 ","End":"03:29.480","Text":"and this one is y^6. That\u0027s a good start."},{"Start":"03:29.480 ","End":"03:31.580","Text":"Now on the rest, we can use the formula."},{"Start":"03:31.580 ","End":"03:34.520","Text":"Now here, k is smaller n minus k,"},{"Start":"03:34.520 ","End":"03:37.385","Text":"so we\u0027ll just put k factorial,"},{"Start":"03:37.385 ","End":"03:40.440","Text":"which is just 1 factorial which is 1,"},{"Start":"03:40.440 ","End":"03:42.990","Text":"and on the numerator, we just start from"},{"Start":"03:42.990 ","End":"03:45.965","Text":"6 and keep filling with the same number of elements."},{"Start":"03:45.965 ","End":"03:50.290","Text":"It\u0027s just 6 over 1, x^5y."},{"Start":"03:50.290 ","End":"03:53.985","Text":"The pattern will be clearer when k is bigger than 1."},{"Start":"03:53.985 ","End":"03:57.420","Text":"Here, for example, 2 is less than 6 minus 2,"},{"Start":"03:57.420 ","End":"03:59.309","Text":"so we start off with 2 factorial,"},{"Start":"03:59.309 ","End":"04:00.900","Text":"which is 2 times 1."},{"Start":"04:00.900 ","End":"04:04.410","Text":"On the numerator, we start from 6 and we count down,"},{"Start":"04:04.410 ","End":"04:06.735","Text":"and match term for term,"},{"Start":"04:06.735 ","End":"04:08.535","Text":"factor for factor, I mean."},{"Start":"04:08.535 ","End":"04:09.960","Text":"Term is when you add factors,"},{"Start":"04:09.960 ","End":"04:13.635","Text":"when you multiply x^4, y^2."},{"Start":"04:13.635 ","End":"04:17.535","Text":"Here, 3 or 6 minus 3, same thing."},{"Start":"04:17.535 ","End":"04:19.560","Text":"It\u0027s going to be 3 factorial,"},{"Start":"04:19.560 ","End":"04:21.180","Text":"which is 3 times 2 times 1,"},{"Start":"04:21.180 ","End":"04:22.575","Text":"and here we start with 6,"},{"Start":"04:22.575 ","End":"04:24.000","Text":"and count down 6,"},{"Start":"04:24.000 ","End":"04:25.260","Text":"5, 4,"},{"Start":"04:25.260 ","End":"04:30.270","Text":"and it will be x^3, y^3."},{"Start":"04:30.270 ","End":"04:33.990","Text":"Then 6 minus 4 is more than 4,"},{"Start":"04:33.990 ","End":"04:36.600","Text":"so we\u0027ll use 2 factorial here."},{"Start":"04:36.600 ","End":"04:40.695","Text":"We\u0027ll get, 2 times 1."},{"Start":"04:40.695 ","End":"04:43.710","Text":"Then from here, 6 times 5,"},{"Start":"04:43.710 ","End":"04:46.755","Text":"and it\u0027s x^2, y^4."},{"Start":"04:46.755 ","End":"04:49.860","Text":"Here, 6 minus 5 is smaller 5,"},{"Start":"04:49.860 ","End":"04:53.370","Text":"so we use the 6 minus 5, is 1 factorial."},{"Start":"04:53.370 ","End":"04:55.275","Text":"Here we just put the 6,"},{"Start":"04:55.275 ","End":"05:00.975","Text":"and we need x, y^5 here."},{"Start":"05:00.975 ","End":"05:04.020","Text":"Now, just the computational part."},{"Start":"05:04.020 ","End":"05:07.200","Text":"Let\u0027s if we got the equals here,"},{"Start":"05:07.200 ","End":"05:09.255","Text":"and the equals here."},{"Start":"05:09.255 ","End":"05:12.705","Text":"So x^6, 6/1 is 6."},{"Start":"05:12.705 ","End":"05:18.090","Text":"6x^5y, 6 times 5 over 2 times 1."},{"Start":"05:18.090 ","End":"05:24.300","Text":"2 goes into 63 times 5 is,15 x^4y^2."},{"Start":"05:24.300 ","End":"05:26.220","Text":"Then here, 3^2 goes with 6,"},{"Start":"05:26.220 ","End":"05:28.780","Text":"so that gives me 20x^3y^3."},{"Start":"05:31.210 ","End":"05:34.010","Text":"Then we get the same thing."},{"Start":"05:34.010 ","End":"05:35.510","Text":"Look, this is the same as this,"},{"Start":"05:35.510 ","End":"05:36.650","Text":"and this is the same as this."},{"Start":"05:36.650 ","End":"05:39.140","Text":"It actually always comes out symmetrical."},{"Start":"05:39.140 ","End":"05:42.470","Text":"Let me just copy the coefficient,15,"},{"Start":"05:42.470 ","End":"05:45.165","Text":"this time it\u0027s x^2y^4."},{"Start":"05:45.165 ","End":"05:52.950","Text":"Then we can copy this as a 6xy^5 plus y^6."},{"Start":"05:52.950 ","End":"05:58.780","Text":"That\u0027s the answer for (x + y)^6,7 terms. Yeah, we\u0027re done."}],"Thumbnail":null,"ID":14125},{"Watched":false,"Name":"Exercise 5","Duration":"6m 44s","ChapterTopicVideoID":13485,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.120","Text":"In this exercise, we have to compute the binomial expansion of (3a-4b)^5."},{"Start":"00:07.120 ","End":"00:09.159","Text":"I\u0027ll write the formula."},{"Start":"00:09.159 ","End":"00:13.325","Text":"I\u0027ll copy it from a previous exercise. Here we are."},{"Start":"00:13.325 ","End":"00:17.625","Text":"N will be 5 and instead of a,"},{"Start":"00:17.625 ","End":"00:20.475","Text":"we will have 3a,"},{"Start":"00:20.475 ","End":"00:22.360","Text":"and wherever we see b,"},{"Start":"00:22.360 ","End":"00:25.780","Text":"we replace that by -4b."},{"Start":"00:25.780 ","End":"00:30.790","Text":"What we\u0027ll get is that this will equal"},{"Start":"00:30.790 ","End":"00:39.030","Text":"the sum k goes from 0 to 5,"},{"Start":"00:39.030 ","End":"00:40.960","Text":"of 5 choose k,"},{"Start":"00:40.960 ","End":"00:45.635","Text":"need the brackets here of course,"},{"Start":"00:45.635 ","End":"00:53.945","Text":"3a^5-k and -4b^k."},{"Start":"00:53.945 ","End":"00:57.590","Text":"This is going to cause an alternation in signs, of course,"},{"Start":"00:57.590 ","End":"01:00.050","Text":"because when you have a minus to an even power,"},{"Start":"01:00.050 ","End":"01:05.404","Text":"it\u0027s 1 plus an odd power it\u0027s going to stay minus."},{"Start":"01:05.404 ","End":"01:07.475","Text":"Let\u0027s see what this comes out."},{"Start":"01:07.475 ","End":"01:09.575","Text":"It will break up the Sigma."},{"Start":"01:09.575 ","End":"01:12.326","Text":"We have 5 over 0,"},{"Start":"01:12.326 ","End":"01:16.395","Text":"they\u0027ll be 6 terms from 0 to 5,"},{"Start":"01:16.395 ","End":"01:22.770","Text":"3a^5 -4b^0."},{"Start":"01:22.770 ","End":"01:26.505","Text":"Then that k equals 1, 5,1,3a."},{"Start":"01:26.505 ","End":"01:34.405","Text":"It\u0027s going to be 4 and these decrease and these increase to the power of 1."},{"Start":"01:34.405 ","End":"01:38.340","Text":"Next, 5 choose 2,"},{"Start":"01:38.340 ","End":"01:43.890","Text":"3a^3 minus 4^2,"},{"Start":"01:44.750 ","End":"01:51.570","Text":"plus continuing the pattern 5,0, 1,2 to 5,3."},{"Start":"01:51.570 ","End":"01:54.645","Text":"5 over k, 3a."},{"Start":"01:54.645 ","End":"01:56.055","Text":"Now, where are we up to?"},{"Start":"01:56.055 ","End":"01:58.395","Text":"3,5,4,3."},{"Start":"01:58.395 ","End":"02:00.195","Text":"This 1 goes down 1,"},{"Start":"02:00.195 ","End":"02:05.340","Text":"the other 1 goes up 1 minus 4b^3."},{"Start":"02:05.340 ","End":"02:08.680","Text":"Next, 5 choose 4,"},{"Start":"02:08.680 ","End":"02:14.780","Text":"3a continuously to decrease down to 1 minus 4b^4,"},{"Start":"02:14.780 ","End":"02:18.140","Text":"and finally, 5 choose 5,"},{"Start":"02:18.140 ","End":"02:28.200","Text":"3a^0, and the minus 4b^5."},{"Start":"02:28.200 ","End":"02:29.750","Text":"Let\u0027s do some computations."},{"Start":"02:29.750 ","End":"02:32.749","Text":"Well, the first and the last we don\u0027t need to compute."},{"Start":"02:32.749 ","End":"02:35.105","Text":"These coefficients are always 1,"},{"Start":"02:35.105 ","End":"02:37.355","Text":"and to the power of 0 we can ignore."},{"Start":"02:37.355 ","End":"02:40.165","Text":"Here we get 3a^5."},{"Start":"02:40.165 ","End":"02:43.400","Text":"I\u0027ll just leave it like that for the moment."},{"Start":"02:43.400 ","End":"02:50.255","Text":"The next one will be 5 choose 1 will use the formula."},{"Start":"02:50.255 ","End":"02:53.210","Text":"You could just get used to the fact that when we"},{"Start":"02:53.210 ","End":"02:56.120","Text":"choose 1 when it\u0027s 1 and the number here,"},{"Start":"02:56.120 ","End":"02:58.235","Text":"it\u0027s just equal to this number above."},{"Start":"02:58.235 ","End":"02:59.660","Text":"If I write it as a product,"},{"Start":"02:59.660 ","End":"03:01.550","Text":"it\u0027s just 5 over 1."},{"Start":"03:01.550 ","End":"03:05.100","Text":"We just take, go from 5 down to just 5."},{"Start":"03:05.100 ","End":"03:09.945","Text":"So well I\u0027ll write it as 5 over 1,1 factorial."},{"Start":"03:09.945 ","End":"03:18.345","Text":"Then we have 3a^4 minus 4b^1 so I won\u0027t bother writing the 1."},{"Start":"03:18.345 ","End":"03:23.040","Text":"Now, 5,2 means we go 2 times 1,"},{"Start":"03:23.040 ","End":"03:30.705","Text":"and then we just go down from 5, 3a^3 minus 4b^2."},{"Start":"03:30.705 ","End":"03:34.635","Text":"Now here, 5 choose 3,"},{"Start":"03:34.635 ","End":"03:39.455","Text":"we can use the formula where there\u0027s a variation where instead of 3,"},{"Start":"03:39.455 ","End":"03:42.335","Text":"we put 5 minus 3 which is 2."},{"Start":"03:42.335 ","End":"03:44.755","Text":"We\u0027ll get the same thing as this."},{"Start":"03:44.755 ","End":"03:47.070","Text":"It\u0027s also 5,4,"},{"Start":"03:47.070 ","End":"03:48.750","Text":"2 times 1,"},{"Start":"03:48.750 ","End":"03:56.490","Text":"and then 3a^2 minus 4b^3."},{"Start":"03:56.490 ","End":"04:00.750","Text":"Then 5,4 is the same as 5,1."},{"Start":"04:00.750 ","End":"04:05.970","Text":"We can always do the 5 minus 4 but these things always come out to be symmetric."},{"Start":"04:05.970 ","End":"04:10.275","Text":"So we get again 5 over 1. This time."},{"Start":"04:10.275 ","End":"04:17.960","Text":"What? 3a and then minus 4b^4 from here then squeeze it in."},{"Start":"04:17.960 ","End":"04:20.180","Text":"Now I have room, the last one,"},{"Start":"04:20.180 ","End":"04:22.330","Text":"the coefficient is always 1,"},{"Start":"04:22.330 ","End":"04:24.570","Text":"and the power of 0 we ignore."},{"Start":"04:24.570 ","End":"04:28.300","Text":"It\u0027s just minus 4b^5."},{"Start":"04:28.430 ","End":"04:31.155","Text":"Continuing."},{"Start":"04:31.155 ","End":"04:40.740","Text":"Let\u0027s compute, 3^5 is 243a^5."},{"Start":"04:40.740 ","End":"04:46.185","Text":"Next, 5 times 3^4th times minus 4,"},{"Start":"04:46.185 ","End":"04:50.550","Text":"it will be minus 3^4 is 81,"},{"Start":"04:50.550 ","End":"04:59.910","Text":"5 times 4 is 20.1620a^4,b."},{"Start":"04:59.910 ","End":"05:02.580","Text":"Next, here we have,"},{"Start":"05:02.580 ","End":"05:05.055","Text":"well let write what these are,"},{"Start":"05:05.055 ","End":"05:09.255","Text":"5 times 4, over 2 times 1 is 10."},{"Start":"05:09.255 ","End":"05:13.290","Text":"That will make this one 10 also."},{"Start":"05:13.290 ","End":"05:15.720","Text":"Yeah, and the rest of them are straightforward."},{"Start":"05:15.720 ","End":"05:18.855","Text":"3^3 is 27,"},{"Start":"05:18.855 ","End":"05:21.765","Text":"4^2 is 16,"},{"Start":"05:21.765 ","End":"05:29.660","Text":"27 times 16 is 432 times 10, makes it 4,320."},{"Start":"05:29.660 ","End":"05:35.750","Text":"It\u0027s a plus because even power here, a^3, b^2."},{"Start":"05:35.750 ","End":"05:47.610","Text":"Then here we have 4^3 is 64 times 9 is 576,"},{"Start":"05:47.610 ","End":"05:51.195","Text":"but it\u0027s also times 10 so let\u0027s take another 0 on,"},{"Start":"05:51.195 ","End":"05:57.115","Text":"and that\u0027s a^2, b^3."},{"Start":"05:57.115 ","End":"06:02.210","Text":"Then 4^4"},{"Start":"06:02.210 ","End":"06:08.650","Text":"is 256 times 15,"},{"Start":"06:08.650 ","End":"06:12.830","Text":"I make that 3,840."},{"Start":"06:12.830 ","End":"06:19.100","Text":"This time it will be minus and it\u0027s ab^4 and"},{"Start":"06:19.100 ","End":"06:26.340","Text":"finally, 4^5 is 1,024b^5."},{"Start":"06:26.340 ","End":"06:28.745","Text":"Let\u0027s get the signs alternate minus plus."},{"Start":"06:28.745 ","End":"06:31.025","Text":"Did something wrong here?"},{"Start":"06:31.025 ","End":"06:33.515","Text":"Minus plus."},{"Start":"06:33.515 ","End":"06:36.695","Text":"Sorry, yeah, the sign should alternate of course."},{"Start":"06:36.695 ","End":"06:39.815","Text":"Minus, plus, minus, plus, minus."},{"Start":"06:39.815 ","End":"06:44.940","Text":"The answer a bit messy but there we go. We\u0027re done."}],"Thumbnail":null,"ID":14126},{"Watched":false,"Name":"Exercise 6","Duration":"3m 2s","ChapterTopicVideoID":13486,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this exercise, we have to write the first 4 terms in"},{"Start":"00:03.750 ","End":"00:08.055","Text":"the binomial expansion of (x+y)^16."},{"Start":"00:08.055 ","End":"00:10.200","Text":"Now you\u0027re supposed to know the formula,"},{"Start":"00:10.200 ","End":"00:11.670","Text":"but in case you\u0027ve forgotten it,"},{"Start":"00:11.670 ","End":"00:13.020","Text":"I\u0027ve written it here."},{"Start":"00:13.020 ","End":"00:20.400","Text":"What we have now is (x+y)^16."},{"Start":"00:20.400 ","End":"00:24.000","Text":"So n is 16 and a and b are x and y."},{"Start":"00:24.000 ","End":"00:28.830","Text":"This is going to equal 16, choose 0,"},{"Start":"00:28.830 ","End":"00:30.960","Text":"x to the power of,"},{"Start":"00:30.960 ","End":"00:36.090","Text":"we started off with the n-k, so that\u0027s 16-0."},{"Start":"00:36.090 ","End":"00:41.700","Text":"That\u0027s 16y^0. Next term,"},{"Start":"00:41.700 ","End":"00:49.680","Text":"16,1, x^16-1, y^1."},{"Start":"00:49.680 ","End":"00:57.900","Text":"Then keep going 16,2 x^14, y^2,"},{"Start":"00:57.900 ","End":"01:06.990","Text":"and 16 choose 3, x^13 y^3."},{"Start":"01:06.990 ","End":"01:09.330","Text":"Now, the first one is always 1,"},{"Start":"01:09.330 ","End":"01:10.814","Text":"and so that\u0027s the easiest,"},{"Start":"01:10.814 ","End":"01:13.915","Text":"y^0 is 1, so it\u0027s x^16."},{"Start":"01:13.915 ","End":"01:16.240","Text":"Now, this we\u0027ve computed enough when it\u0027s over 1,"},{"Start":"01:16.240 ","End":"01:19.600","Text":"it\u0027s just 16, don\u0027t bother with expanding it."},{"Start":"01:19.600 ","End":"01:24.035","Text":"x^15, y^1 is y."},{"Start":"01:24.035 ","End":"01:29.300","Text":"Now 16 choose 2 is 16 to the denominator first,"},{"Start":"01:29.300 ","End":"01:38.135","Text":"is 2!,16*15, same number of factors, x^14, y^2."},{"Start":"01:38.135 ","End":"01:40.315","Text":"Then here we have 3!"},{"Start":"01:40.315 ","End":"01:41.945","Text":"on the bottom."},{"Start":"01:41.945 ","End":"01:43.595","Text":"Or we could have done 13!"},{"Start":"01:43.595 ","End":"01:45.019","Text":"Whichever is smaller,"},{"Start":"01:45.019 ","End":"01:49.130","Text":"and then 16, 15, 14."},{"Start":"01:49.130 ","End":"01:50.690","Text":"You don\u0027t have to do any computations here,"},{"Start":"01:50.690 ","End":"01:52.820","Text":"you just have to make sure that 3, 2,"},{"Start":"01:52.820 ","End":"01:55.975","Text":"1 are the 3 factors here,"},{"Start":"01:55.975 ","End":"01:59.720","Text":"x^13, y^3,"},{"Start":"01:59.720 ","End":"02:01.790","Text":"and then just numerical computation."},{"Start":"02:01.790 ","End":"02:08.645","Text":"So x^16 here I\u0027m copying 16x^15, y,"},{"Start":"02:08.645 ","End":"02:11.210","Text":"2 into 16 goes 8 times,"},{"Start":"02:11.210 ","End":"02:19.050","Text":"8*15 is 120x^14, y^2."},{"Start":"02:19.050 ","End":"02:24.420","Text":"Then let\u0027s see, 3 goes into 15,"},{"Start":"02:24.420 ","End":"02:26.460","Text":"so that\u0027s going to be 5."},{"Start":"02:26.460 ","End":"02:29.940","Text":"Then 2 into 16 or into 14,"},{"Start":"02:29.940 ","End":"02:33.314","Text":"doesn\u0027t matter, go into 16, 8 times."},{"Start":"02:33.314 ","End":"02:36.510","Text":"Now we need to compute 8*5*14,"},{"Start":"02:36.510 ","End":"02:44.460","Text":"that\u0027s it\u0027s 40*14, 4*14=56."},{"Start":"02:44.460 ","End":"02:47.820","Text":"That\u0027s 560x^13, y^3."},{"Start":"02:47.820 ","End":"02:50.405","Text":"Let\u0027s just the first 4 terms."},{"Start":"02:50.405 ","End":"02:54.545","Text":"You should really indicate that this goes on and so on."},{"Start":"02:54.545 ","End":"02:58.565","Text":"In fact, we even know the last term will be y^16."},{"Start":"02:58.565 ","End":"03:02.880","Text":"We don\u0027t have to write that. I\u0027m just putting it in there and that\u0027s it."}],"Thumbnail":null,"ID":14127},{"Watched":false,"Name":"Exercise 7","Duration":"3m 25s","ChapterTopicVideoID":13487,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have to write just the first 3 terms in"},{"Start":"00:03.270 ","End":"00:09.195","Text":"the binomial expansion of this plus this to the 10th and here\u0027s the formula."},{"Start":"00:09.195 ","End":"00:12.735","Text":"This will be a, this will be b,"},{"Start":"00:12.735 ","End":"00:17.025","Text":"and this will be n. Let\u0027s see."},{"Start":"00:17.025 ","End":"00:20.355","Text":"What we get for this, I\u0027ll copy it,"},{"Start":"00:20.355 ","End":"00:25.590","Text":"and what we get is 10 choose 0,"},{"Start":"00:25.590 ","End":"00:30.750","Text":"the first one 3u^2 to the power of"},{"Start":"00:30.750 ","End":"00:36.885","Text":"10 and 2v^3 to the power of 0 and keep going."},{"Start":"00:36.885 ","End":"00:45.260","Text":"It\u0027s 10 this time it\u0027s 1 3u^2 minus 1 and to the cube,"},{"Start":"00:45.260 ","End":"00:46.955","Text":"to the power of 1."},{"Start":"00:46.955 ","End":"00:51.080","Text":"And the third term, 10 choose 2,"},{"Start":"00:51.080 ","End":"00:54.120","Text":"3u^2 to the 10 minus 2,"},{"Start":"00:54.120 ","End":"00:57.870","Text":"2v^3 to the power of 2."},{"Start":"00:57.870 ","End":"00:59.820","Text":"Look it\u0027s the first 3 terms,"},{"Start":"00:59.820 ","End":"01:02.035","Text":"so I\u0027ll write plus dot dot dot."},{"Start":"01:02.035 ","End":"01:03.935","Text":"I don\u0027t mean that this is equal to this."},{"Start":"01:03.935 ","End":"01:05.480","Text":"What does this equal to?"},{"Start":"01:05.480 ","End":"01:07.520","Text":"Well, the first term is always the easiest."},{"Start":"01:07.520 ","End":"01:10.910","Text":"It\u0027s just 3u^2 to the 10th."},{"Start":"01:10.910 ","End":"01:16.130","Text":"It\u0027s in general, a^n plus 10 choose 1."},{"Start":"01:16.130 ","End":"01:18.619","Text":"We don\u0027t need to use the formula."},{"Start":"01:18.619 ","End":"01:20.720","Text":"N over 1 is just n,"},{"Start":"01:20.720 ","End":"01:25.685","Text":"so that\u0027s 10 times 3u^2 to the 9th,"},{"Start":"01:25.685 ","End":"01:31.455","Text":"2v^3 and then here we will use the formula."},{"Start":"01:31.455 ","End":"01:35.960","Text":"We can put on the bottom either k factorial or n minus k factorial,"},{"Start":"01:35.960 ","End":"01:39.380","Text":"whichever is smaller so that\u0027s 2 factorial, 2 times 1."},{"Start":"01:39.380 ","End":"01:44.420","Text":"Then the rule is just start with 10 and go down matching factor for factor."},{"Start":"01:44.420 ","End":"01:48.170","Text":"Technically this would be up to 10 minus 2 plus 1,"},{"Start":"01:48.170 ","End":"01:53.255","Text":"which is 9 and then 3u^2 to the a,"},{"Start":"01:53.255 ","End":"01:58.505","Text":"2v^3 squared plus dot dot dot equals,"},{"Start":"01:58.505 ","End":"02:01.460","Text":"I don\u0027t know that we have to compute 3^10."},{"Start":"02:01.460 ","End":"02:02.915","Text":"If we don\u0027t have a calculator,"},{"Start":"02:02.915 ","End":"02:05.885","Text":"you could leave it as 3^10."},{"Start":"02:05.885 ","End":"02:11.165","Text":"Then u^2 to the 10th would be u^20 plus,"},{"Start":"02:11.165 ","End":"02:12.350","Text":"now 10 times 2,"},{"Start":"02:12.350 ","End":"02:14.900","Text":"I can multiply without a calculator."},{"Start":"02:14.900 ","End":"02:18.860","Text":"We\u0027ll write 20 and the 3^9."},{"Start":"02:18.860 ","End":"02:21.350","Text":"If you want to actually compute the number,"},{"Start":"02:21.350 ","End":"02:22.595","Text":"I\u0027ll leave it like that."},{"Start":"02:22.595 ","End":"02:28.585","Text":"Then we have u^2 the ninth is u^18, and then v^3."},{"Start":"02:28.585 ","End":"02:30.830","Text":"The next one will be,"},{"Start":"02:30.830 ","End":"02:33.694","Text":"let\u0027s see what we can compute this."},{"Start":"02:33.694 ","End":"02:35.795","Text":"I\u0027ll do that at the side or maybe here,"},{"Start":"02:35.795 ","End":"02:40.695","Text":"10 times 9 over 2 times 1 the coefficient."},{"Start":"02:40.695 ","End":"02:45.430","Text":"Then we\u0027ll need 3^8 and then 2^2."},{"Start":"02:45.430 ","End":"02:48.785","Text":"Well, some of it we can do without the calculator."},{"Start":"02:48.785 ","End":"02:54.500","Text":"This 2 can cancel with the channel or better still cancel it with this 2."},{"Start":"02:54.500 ","End":"02:55.970","Text":"What do we have here?"},{"Start":"02:55.970 ","End":"03:03.180","Text":"10 times 9 times 2,180. I\u0027ll write it here,180."},{"Start":"03:03.430 ","End":"03:08.870","Text":"Then 3^8 is a bit big to compute without a calculator,"},{"Start":"03:08.870 ","End":"03:11.465","Text":"so I\u0027ll just write it as 3^8."},{"Start":"03:11.465 ","End":"03:15.530","Text":"Now, u^2 to the 8 is u^16,"},{"Start":"03:15.530 ","End":"03:20.780","Text":"and v^3 squared is v^6 plus dot dot dot,"},{"Start":"03:20.780 ","End":"03:25.710","Text":"because this is just the first 3 terms. That\u0027s it."}],"Thumbnail":null,"ID":14128},{"Watched":false,"Name":"Exercise 8","Duration":"2m 39s","ChapterTopicVideoID":13488,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.825","Text":"In this exercise, we\u0027re asked to prove a formula relating to the binomial coefficients."},{"Start":"00:06.825 ","End":"00:10.110","Text":"The formula basically says what\u0027s written here,"},{"Start":"00:10.110 ","End":"00:16.725","Text":"that n choose k is the same as n choose n minus k. Before we get into prove,"},{"Start":"00:16.725 ","End":"00:20.100","Text":"I\u0027d like to give an example from 1 of the earlier exercises,"},{"Start":"00:20.100 ","End":"00:26.295","Text":"where we computed all the binomial coefficients relating to n equals 6."},{"Start":"00:26.295 ","End":"00:30.000","Text":"We computed these, I\u0027ll just write in this form,"},{"Start":"00:30.000 ","End":"00:33.510","Text":"and then I\u0027ll give the numerical results that we got,"},{"Start":"00:33.510 ","End":"00:35.355","Text":"6, 5,"},{"Start":"00:35.355 ","End":"00:37.515","Text":"and finally 6, 6."},{"Start":"00:37.515 ","End":"00:41.100","Text":"Now the answers we got were 1 here,"},{"Start":"00:41.100 ","End":"00:43.995","Text":"6, 15,"},{"Start":"00:43.995 ","End":"00:48.090","Text":"this was 20, this was 15,"},{"Start":"00:48.090 ","End":"00:49.665","Text":"this was 6,"},{"Start":"00:49.665 ","End":"00:51.015","Text":"and this was 1."},{"Start":"00:51.015 ","End":"00:53.950","Text":"Now, notice that there\u0027s asymmetry."},{"Start":"00:53.950 ","End":"00:56.724","Text":"This is equal to this,"},{"Start":"00:56.724 ","End":"00:59.410","Text":"and this is equal to this."},{"Start":"00:59.410 ","End":"01:01.205","Text":"This is equal to this."},{"Start":"01:01.205 ","End":"01:03.005","Text":"Well, this is equal to itself."},{"Start":"01:03.005 ","End":"01:04.430","Text":"Now, if you look at them,"},{"Start":"01:04.430 ","End":"01:06.890","Text":"this means that 6, 0,"},{"Start":"01:06.890 ","End":"01:09.485","Text":"and 6, 6 are the same."},{"Start":"01:09.485 ","End":"01:12.995","Text":"If I take 6 and subtract the 0,"},{"Start":"01:12.995 ","End":"01:14.315","Text":"I get the 6 here."},{"Start":"01:14.315 ","End":"01:15.980","Text":"6 minus 1 is 5,"},{"Start":"01:15.980 ","End":"01:17.540","Text":"or 6 minus 5 is 1."},{"Start":"01:17.540 ","End":"01:20.930","Text":"In other words, if I subtract the lower 1 from the upper 1,"},{"Start":"01:20.930 ","End":"01:22.400","Text":"and put it here,"},{"Start":"01:22.400 ","End":"01:24.370","Text":"you get the same thing, 6,"},{"Start":"01:24.370 ","End":"01:27.010","Text":"2 and 6, 4 are the same."},{"Start":"01:27.010 ","End":"01:28.420","Text":"Mark that here."},{"Start":"01:28.420 ","End":"01:29.945","Text":"This is the same as this,"},{"Start":"01:29.945 ","End":"01:31.708","Text":"this is the same as this,"},{"Start":"01:31.708 ","End":"01:35.095","Text":"and this is itself because 6 minus 3 is 3."},{"Start":"01:35.095 ","End":"01:39.245","Text":"So 6 minus 4 is 2 and so on. You get the idea."},{"Start":"01:39.245 ","End":"01:40.730","Text":"Now let\u0027s prove it."},{"Start":"01:40.730 ","End":"01:44.105","Text":"Here goes. I\u0027ll start with this,"},{"Start":"01:44.105 ","End":"01:45.695","Text":"and I\u0027ll reach this."},{"Start":"01:45.695 ","End":"01:51.470","Text":"n choose n minus k is equal to what\u0027s upstairs"},{"Start":"01:51.470 ","End":"01:59.705","Text":"factorial over what\u0027s downstairs factorial times upstairs minus downstairs,"},{"Start":"01:59.705 ","End":"02:04.805","Text":"n minus n minus k factorial."},{"Start":"02:04.805 ","End":"02:10.310","Text":"Now, n minus n minus k is just k. What we get"},{"Start":"02:10.310 ","End":"02:16.250","Text":"is m factorial over n minus k factorial,"},{"Start":"02:16.250 ","End":"02:18.725","Text":"and then k factorial."},{"Start":"02:18.725 ","End":"02:21.860","Text":"Now, I can change the order of things."},{"Start":"02:21.860 ","End":"02:23.495","Text":"We\u0027re almost there."},{"Start":"02:23.495 ","End":"02:27.365","Text":"All I have to do is write k factorial,"},{"Start":"02:27.365 ","End":"02:29.780","Text":"and then n minus k factorial."},{"Start":"02:29.780 ","End":"02:31.775","Text":"If you go look at the definition,"},{"Start":"02:31.775 ","End":"02:36.605","Text":"it\u0027s exactly the definition of n choose k. So this equals this,"},{"Start":"02:36.605 ","End":"02:40.320","Text":"which is what we have to prove. We\u0027re done."}],"Thumbnail":null,"ID":14129},{"Watched":false,"Name":"Exercise 9","Duration":"6m 46s","ChapterTopicVideoID":13489,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:07.395","Text":"In this exercise, we have to prove an identity involving the binomial coefficients."},{"Start":"00:07.395 ","End":"00:09.360","Text":"I won\u0027t read it out, it\u0027s as written."},{"Start":"00:09.360 ","End":"00:11.460","Text":"But let\u0027s take an example first."},{"Start":"00:11.460 ","End":"00:18.390","Text":"Suppose I take n=5 and k=2."},{"Start":"00:18.390 ","End":"00:22.785","Text":"Then what I would get is 5 choose,"},{"Start":"00:22.785 ","End":"00:24.840","Text":"k minus 1 is 1,"},{"Start":"00:24.840 ","End":"00:28.140","Text":"and then k itself is 2."},{"Start":"00:28.140 ","End":"00:30.810","Text":"Well, we don\u0027t know this is what the claim is,"},{"Start":"00:30.810 ","End":"00:32.580","Text":"would equal n plus 1,"},{"Start":"00:32.580 ","End":"00:35.505","Text":"which is 6 choose 2."},{"Start":"00:35.505 ","End":"00:42.060","Text":"Now, let\u0027s verify that 5 choose 1 is 5,"},{"Start":"00:42.060 ","End":"00:48.320","Text":"5 choose 2 is 2 factorial on the denominator then 5 times"},{"Start":"00:48.320 ","End":"00:55.935","Text":"4 equals question mark 2 times 1, 6 times 5."},{"Start":"00:55.935 ","End":"00:57.555","Text":"What are we saying?"},{"Start":"00:57.555 ","End":"01:02.655","Text":"That 5 plus this comes out to be 10,"},{"Start":"01:02.655 ","End":"01:04.350","Text":"and this comes out to be,"},{"Start":"01:04.350 ","End":"01:05.490","Text":"2 into 6, 3,"},{"Start":"01:05.490 ","End":"01:07.380","Text":"3 times 5 is 15."},{"Start":"01:07.380 ","End":"01:10.095","Text":"Well, that\u0027s certainly true."},{"Start":"01:10.095 ","End":"01:14.760","Text":"Let\u0027s go and prove it and we\u0027ll use the definition."},{"Start":"01:14.760 ","End":"01:17.595","Text":"Start with the left-hand side,"},{"Start":"01:17.595 ","End":"01:25.970","Text":"and what we have n choose k minus 1 is n factorial over."},{"Start":"01:25.970 ","End":"01:28.370","Text":"Now this thing factorial,"},{"Start":"01:28.370 ","End":"01:36.690","Text":"which is k minus 1 factorial and then n minus k minus 1 factorial and subtract k minus 1."},{"Start":"01:36.690 ","End":"01:41.235","Text":"I\u0027ll do it as n minus k plus 1 factorial."},{"Start":"01:41.235 ","End":"01:47.360","Text":"The other one is n factorial over k factorial,"},{"Start":"01:47.360 ","End":"01:51.335","Text":"n minus k factorial straight from the definition."},{"Start":"01:51.335 ","End":"01:53.664","Text":"Now that\u0027s the left-hand side."},{"Start":"01:53.664 ","End":"01:57.560","Text":"What we\u0027ll do is we\u0027ll start with this and we\u0027ll see if we can"},{"Start":"01:57.560 ","End":"02:03.320","Text":"reach the left-hand side, right-hand side."},{"Start":"02:03.320 ","End":"02:05.450","Text":"That\u0027s one way of showing equality,"},{"Start":"02:05.450 ","End":"02:09.545","Text":"stuff on one side and keep developing it until we get to the other side."},{"Start":"02:09.545 ","End":"02:12.575","Text":"Let\u0027s see what we can do to simplify this."},{"Start":"02:12.575 ","End":"02:14.960","Text":"Now here we have k minus 1 factorial."},{"Start":"02:14.960 ","End":"02:16.865","Text":"Here we have k factorial."},{"Start":"02:16.865 ","End":"02:18.860","Text":"I could say in general,"},{"Start":"02:18.860 ","End":"02:24.170","Text":"that some number r factorial is"},{"Start":"02:24.170 ","End":"02:30.140","Text":"equal to r times r minus 1 factorial."},{"Start":"02:30.140 ","End":"02:35.420","Text":"Because here it\u0027s all the numbers from r down to 1 multiplied together."},{"Start":"02:35.420 ","End":"02:38.190","Text":"If I take the first factor is r,"},{"Start":"02:38.190 ","End":"02:40.950","Text":"and I\u0027ll get from r minus 1 down to 1."},{"Start":"02:40.950 ","End":"02:45.990","Text":"For example, 6 factorial is 6 times 5 factorial."},{"Start":"02:45.990 ","End":"02:48.630","Text":"If you write down from 6 down to 1,"},{"Start":"02:48.630 ","End":"02:51.360","Text":"just take the 6th and we\u0027ll go from 5 down to 1."},{"Start":"02:51.360 ","End":"02:54.950","Text":"I\u0027m going to work on both of these simultaneously."},{"Start":"02:54.950 ","End":"02:57.965","Text":"Here n factorial, here n factorial."},{"Start":"02:57.965 ","End":"03:00.770","Text":"Now, if I take r to be k,"},{"Start":"03:00.770 ","End":"03:02.780","Text":"which is what I\u0027ve set out to do."},{"Start":"03:02.780 ","End":"03:06.545","Text":"Then here we\u0027ll have k minus 1 factorial,"},{"Start":"03:06.545 ","End":"03:13.225","Text":"but here instead of k factorial I\u0027ll write k times k minus 1 factorial."},{"Start":"03:13.225 ","End":"03:16.640","Text":"Now, these two are starting to look closer to each other."},{"Start":"03:16.640 ","End":"03:18.815","Text":"I\u0027m going to use this rule again."},{"Start":"03:18.815 ","End":"03:28.860","Text":"I can write this as n minus k plus 1 times this thing without the plus 1,"},{"Start":"03:28.860 ","End":"03:36.945","Text":"because this is like r and I\u0027ll take r minus 1 will be just n minus k factorial."},{"Start":"03:36.945 ","End":"03:42.515","Text":"Here, just copy the n minus k factorial."},{"Start":"03:42.515 ","End":"03:46.175","Text":"Now, these are very similar."},{"Start":"03:46.175 ","End":"03:53.540","Text":"The differences being that here we have n minus k plus 1,"},{"Start":"03:53.540 ","End":"03:57.410","Text":"and here we have k. But other than that,"},{"Start":"03:57.410 ","End":"04:00.520","Text":"both these expressions are the same."},{"Start":"04:00.520 ","End":"04:04.385","Text":"What we get if I take the common part out,"},{"Start":"04:04.385 ","End":"04:12.230","Text":"which is n factorial over k minus 1 factorial,"},{"Start":"04:12.230 ","End":"04:16.260","Text":"n minus k factorial."},{"Start":"04:16.260 ","End":"04:23.280","Text":"Then we have 1 over n minus k plus"},{"Start":"04:23.280 ","End":"04:30.575","Text":"1 plus 1 over k. What we get,"},{"Start":"04:30.575 ","End":"04:37.835","Text":"now copy this n factorial over k minus 1 factorial,"},{"Start":"04:37.835 ","End":"04:40.460","Text":"n minus k factorial."},{"Start":"04:40.460 ","End":"04:42.745","Text":"Now if we add these,"},{"Start":"04:42.745 ","End":"04:45.725","Text":"if we put it over a common denominator,"},{"Start":"04:45.725 ","End":"04:47.930","Text":"which is this times this,"},{"Start":"04:47.930 ","End":"04:53.820","Text":"what we\u0027ll get is n minus k plus 1 and"},{"Start":"04:53.820 ","End":"04:59.610","Text":"then k. Then we\u0027ll get k plus n minus k plus 1."},{"Start":"04:59.610 ","End":"05:01.890","Text":"If we think about it,"},{"Start":"05:01.890 ","End":"05:06.855","Text":"k plus n minus k plus 1 is just n plus 1."},{"Start":"05:06.855 ","End":"05:09.270","Text":"Here we have n plus 1."},{"Start":"05:09.270 ","End":"05:11.705","Text":"Now if we use this rule again."},{"Start":"05:11.705 ","End":"05:15.245","Text":"In fact, we\u0027ll use this rule three times."},{"Start":"05:15.245 ","End":"05:19.550","Text":"First of all, if I take r to be n plus 1,"},{"Start":"05:19.550 ","End":"05:20.840","Text":"then these two,"},{"Start":"05:20.840 ","End":"05:26.840","Text":"the numerator combines to equal n plus 1 factorial."},{"Start":"05:26.840 ","End":"05:28.310","Text":"That\u0027s what this and this."},{"Start":"05:28.310 ","End":"05:33.485","Text":"Then if I take the k minus 1 factorial with the k,"},{"Start":"05:33.485 ","End":"05:38.660","Text":"that will give me k factorial that\u0027s on the denominator,"},{"Start":"05:38.660 ","End":"05:41.405","Text":"and perhaps I should indicate,"},{"Start":"05:41.405 ","End":"05:43.535","Text":"we\u0027re taking this with this,"},{"Start":"05:43.535 ","End":"05:46.774","Text":"with this rule, this with this."},{"Start":"05:46.774 ","End":"05:49.670","Text":"Then again, this with this."},{"Start":"05:49.670 ","End":"05:52.790","Text":"The one we haven\u0027t done is this with this and not just"},{"Start":"05:52.790 ","End":"05:56.240","Text":"becomes n minus k plus 1 factorial."},{"Start":"05:56.240 ","End":"05:59.120","Text":"Now instead of n minus k plus 1,"},{"Start":"05:59.120 ","End":"06:02.590","Text":"well, I\u0027ll just indicate."},{"Start":"06:02.590 ","End":"06:06.530","Text":"But this is obviously the same as if I wrote n plus"},{"Start":"06:06.530 ","End":"06:10.595","Text":"1 minus k. The reason I want to do that"},{"Start":"06:10.595 ","End":"06:19.185","Text":"is that then this comes out exactly to be the definition of n plus 1 choose k,"},{"Start":"06:19.185 ","End":"06:21.935","Text":"because here\u0027s the n plus 1 factorial,"},{"Start":"06:21.935 ","End":"06:23.735","Text":"here\u0027s the k factorial,"},{"Start":"06:23.735 ","End":"06:29.510","Text":"and this minus this is n plus 1 minus k. We have"},{"Start":"06:29.510 ","End":"06:35.599","Text":"exactly arrived from the left-hand side to the right-hand side."},{"Start":"06:35.599 ","End":"06:36.905","Text":"We started from here,"},{"Start":"06:36.905 ","End":"06:38.375","Text":"we got this and this,"},{"Start":"06:38.375 ","End":"06:40.805","Text":"developed it and we got here."},{"Start":"06:40.805 ","End":"06:43.074","Text":"As they say in Latin,"},{"Start":"06:43.074 ","End":"06:46.810","Text":"QED we proved, what had to be proved."}],"Thumbnail":null,"ID":14130},{"Watched":false,"Name":"Exercise 10","Duration":"5m 1s","ChapterTopicVideoID":13490,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.090","Text":"This exercise is really best explained with diagrams rather than in words."},{"Start":"00:06.090 ","End":"00:09.270","Text":"It\u0027s hard to follow from a few words what it means."},{"Start":"00:09.270 ","End":"00:13.875","Text":"But basically, if you arrange the binomial into a triangle form,"},{"Start":"00:13.875 ","End":"00:20.535","Text":"centered symmetrically, then like a plus b^2 is a^2 plus 2ab plus b^2."},{"Start":"00:20.535 ","End":"00:22.710","Text":"We take the coefficients 121."},{"Start":"00:22.710 ","End":"00:26.250","Text":"When going to keep going we get what is called Pascal\u0027s triangle,"},{"Start":"00:26.250 ","End":"00:28.605","Text":"which is really infinite. It goes on forever."},{"Start":"00:28.605 ","End":"00:33.165","Text":"Let me explain what we want here with a couple of diagrams."},{"Start":"00:33.165 ","End":"00:37.240","Text":"For example, this last row, 14641."},{"Start":"00:37.240 ","End":"00:41.520","Text":"These are the coefficients of a plus b^4."},{"Start":"00:41.770 ","End":"00:44.375","Text":"This is row 4,"},{"Start":"00:44.375 ","End":"00:48.900","Text":"so it\u0027s 4 choose 0 is the co-efficient,"},{"Start":"00:48.900 ","End":"00:50.820","Text":"the first term, then 4 choose 1,"},{"Start":"00:50.820 ","End":"00:53.415","Text":"4 choose 2, 4,3, 4,4."},{"Start":"00:53.415 ","End":"00:56.060","Text":"We know that the first and last ones are always 1."},{"Start":"00:56.060 ","End":"01:00.125","Text":"If we make the computations say up to 10,"},{"Start":"01:00.125 ","End":"01:02.930","Text":"then here\u0027s what it comes out numerically."},{"Start":"01:02.930 ","End":"01:07.910","Text":"The point is that there\u0027s a pattern that enables us to compute these very quickly."},{"Start":"01:07.910 ","End":"01:11.960","Text":"But more than that, we don\u0027t even have to compute them if you\u0027d like print out"},{"Start":"01:11.960 ","End":"01:15.860","Text":"or keep as an image up to 10 or up to however many you\u0027ll need,"},{"Start":"01:15.860 ","End":"01:17.750","Text":"then you don\u0027t have to actually do computations,"},{"Start":"01:17.750 ","End":"01:19.490","Text":"you can just refer to the diagram."},{"Start":"01:19.490 ","End":"01:21.860","Text":"But if you do want to generate the triangle,"},{"Start":"01:21.860 ","End":"01:23.630","Text":"then there\u0027s a simple rule."},{"Start":"01:23.630 ","End":"01:26.825","Text":"Let\u0027s say we\u0027ve computed it up to 4."},{"Start":"01:26.825 ","End":"01:29.180","Text":"We have up to here,"},{"Start":"01:29.180 ","End":"01:31.040","Text":"then we don\u0027t have 5 onwards."},{"Start":"01:31.040 ","End":"01:33.950","Text":"What we would do is first of all, for the next row,"},{"Start":"01:33.950 ","End":"01:36.965","Text":"we place a 1 at the beginning and at the end,"},{"Start":"01:36.965 ","End":"01:39.230","Text":"and then each entry,"},{"Start":"01:39.230 ","End":"01:40.400","Text":"this will be a blank,"},{"Start":"01:40.400 ","End":"01:43.775","Text":"but I\u0027d say 1 plus 4 and get 5,"},{"Start":"01:43.775 ","End":"01:45.050","Text":"4 plus 6,"},{"Start":"01:45.050 ","End":"01:47.825","Text":"I get 10, 6 plus 4, 10."},{"Start":"01:47.825 ","End":"01:49.400","Text":"4 plus 1, 5."},{"Start":"01:49.400 ","End":"01:52.040","Text":"We put a 1 at the beginning and 1 at the end."},{"Start":"01:52.040 ","End":"01:53.615","Text":"Then we\u0027ve got the fifth row,"},{"Start":"01:53.615 ","End":"01:54.860","Text":"then the sixth row,"},{"Start":"01:54.860 ","End":"01:56.480","Text":"1 at the beginning, 1 at the end."},{"Start":"01:56.480 ","End":"01:58.915","Text":"Then we would compute it by 1 plus 5."},{"Start":"01:58.915 ","End":"02:00.840","Text":"10 plus 5 is 15."},{"Start":"02:00.840 ","End":"02:03.915","Text":"10 and 10 is 20, and so on."},{"Start":"02:03.915 ","End":"02:05.750","Text":"It looks like this rule works."},{"Start":"02:05.750 ","End":"02:07.639","Text":"How would we describe this in general?"},{"Start":"02:07.639 ","End":"02:10.895","Text":"Let\u0027s look at a specific example."},{"Start":"02:10.895 ","End":"02:13.940","Text":"Let\u0027s go to the next row is 35,"},{"Start":"02:13.940 ","End":"02:17.270","Text":"which we compute from 15 plus 20."},{"Start":"02:17.270 ","End":"02:20.615","Text":"What is that in terms of the binomial coefficients?"},{"Start":"02:20.615 ","End":"02:27.121","Text":"It\u0027s the fact that this plus this is equal to this,"},{"Start":"02:27.121 ","End":"02:29.105","Text":"because this is 7 choose 3;"},{"Start":"02:29.105 ","End":"02:30.905","Text":"0, 1, 2, 3."},{"Start":"02:30.905 ","End":"02:33.350","Text":"What is this? This is 6 choose 0,"},{"Start":"02:33.350 ","End":"02:35.060","Text":"1, 2, and 3."},{"Start":"02:35.060 ","End":"02:37.910","Text":"It\u0027s 6,2 plus 6,3 is 7,3."},{"Start":"02:37.910 ","End":"02:47.100","Text":"I\u0027ll write that example, 6,2 plus 6,3=7,3."},{"Start":"02:47.100 ","End":"02:49.145","Text":"If we generalize this,"},{"Start":"02:49.145 ","End":"02:53.314","Text":"notice that when you go upwards into the right,"},{"Start":"02:53.314 ","End":"02:56.510","Text":"the number below is the same."},{"Start":"02:56.510 ","End":"03:02.295","Text":"Like here, all the entries have a 3 here."},{"Start":"03:02.295 ","End":"03:09.545","Text":"If we go up, we reduce the top number by 1 and keep the bottom number like we did here."},{"Start":"03:09.545 ","End":"03:11.720","Text":"We increase for the right one,"},{"Start":"03:11.720 ","End":"03:13.955","Text":"the 6, we increased by 1 and leave that."},{"Start":"03:13.955 ","End":"03:15.590","Text":"Then for the other entry,"},{"Start":"03:15.590 ","End":"03:17.390","Text":"we just subtract 1 from here."},{"Start":"03:17.390 ","End":"03:19.310","Text":"What we would get in general,"},{"Start":"03:19.310 ","End":"03:22.820","Text":"and what we\u0027re required to prove is the following;"},{"Start":"03:22.820 ","End":"03:32.895","Text":"n choose k plus n choose k plus 1 will equal n plus 1,"},{"Start":"03:32.895 ","End":"03:35.940","Text":"choose k plus 1."},{"Start":"03:35.940 ","End":"03:40.660","Text":"But here k has to be less than n. Well, bigger or equal to 0,"},{"Start":"03:40.660 ","End":"03:45.535","Text":"of course, and less than n. Otherwise we can\u0027t do the k plus 1."},{"Start":"03:45.535 ","End":"03:49.260","Text":"In any event, k will be usually bigger than 0 also."},{"Start":"03:49.260 ","End":"03:51.340","Text":"We\u0027re not interested in the ones at the end."},{"Start":"03:51.340 ","End":"03:53.125","Text":"How do we prove this?"},{"Start":"03:53.125 ","End":"03:55.150","Text":"We use the formula, but wait,"},{"Start":"03:55.150 ","End":"03:59.170","Text":"we don\u0027t have to actually prove it because it\u0027s a previous exercise where we"},{"Start":"03:59.170 ","End":"04:03.640","Text":"showed this identity of the formula, but not quite."},{"Start":"04:03.640 ","End":"04:04.900","Text":"If I instead of k,"},{"Start":"04:04.900 ","End":"04:07.195","Text":"put k minus 1,"},{"Start":"04:07.195 ","End":"04:10.060","Text":"it\u0027s not clear which to choose this k. The 3 or the 2 here."},{"Start":"04:10.060 ","End":"04:11.815","Text":"If we chose the 3,"},{"Start":"04:11.815 ","End":"04:14.140","Text":"then we would express this slightly differently."},{"Start":"04:14.140 ","End":"04:19.570","Text":"We would say n choose k minus 1 plus n choose"},{"Start":"04:19.570 ","End":"04:25.070","Text":"k is equal to n plus 1 choose k. In this case,"},{"Start":"04:25.070 ","End":"04:28.730","Text":"the condition would be that k is bigger than 0"},{"Start":"04:28.730 ","End":"04:32.765","Text":"and k has to be less than or equal to n. This,"},{"Start":"04:32.765 ","End":"04:38.210","Text":"if you check back that we solved this in a previous exercise."},{"Start":"04:38.210 ","End":"04:40.010","Text":"Actually we\u0027re done."},{"Start":"04:40.010 ","End":"04:42.135","Text":"There\u0027s nothing more to do."},{"Start":"04:42.135 ","End":"04:43.820","Text":"Those who didn\u0027t know it,"},{"Start":"04:43.820 ","End":"04:45.320","Text":"this is Pascal\u0027s triangle."},{"Start":"04:45.320 ","End":"04:47.570","Text":"I recommend printing out,"},{"Start":"04:47.570 ","End":"04:49.460","Text":"let\u0027s say up to whatever,"},{"Start":"04:49.460 ","End":"04:51.545","Text":"up to 10 or however many."},{"Start":"04:51.545 ","End":"04:53.510","Text":"Then you don\u0027t have to do computations."},{"Start":"04:53.510 ","End":"04:56.930","Text":"Well, you\u0027d have to ask the instructor if you\u0027re allowed to bring"},{"Start":"04:56.930 ","End":"05:01.590","Text":"in Pascal\u0027s triangle. That\u0027s it for this clip."}],"Thumbnail":null,"ID":14131},{"Watched":false,"Name":"Exercise 11","Duration":"12m 29s","ChapterTopicVideoID":13491,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.465","Text":"This is a more theoretical exercise."},{"Start":"00:03.465 ","End":"00:07.170","Text":"So far we\u0027ve been using Newton\u0027s binomial theorem,"},{"Start":"00:07.170 ","End":"00:08.730","Text":"but we haven\u0027t proved it."},{"Start":"00:08.730 ","End":"00:12.780","Text":"In this exercise, we\u0027re asked to prove it using Mathematical Induction."},{"Start":"00:12.780 ","End":"00:17.010","Text":"I\u0027m going to assume that you know what mathematical induction is."},{"Start":"00:17.010 ","End":"00:22.920","Text":"Essentially, what we have to show with induction is that first of all,"},{"Start":"00:22.920 ","End":"00:27.030","Text":"it\u0027s true for n=1, that it\u0027s true,"},{"Start":"00:27.030 ","End":"00:33.525","Text":"and then we have to make sure that if it\u0027s true for a particular n,"},{"Start":"00:33.525 ","End":"00:38.180","Text":"then it\u0027s going to be true for n plus 1,"},{"Start":"00:38.180 ","End":"00:42.030","Text":"or I\u0027d rather replace n by k. I\u0027ll rephrase that."},{"Start":"00:42.030 ","End":"00:45.755","Text":"If it\u0027s true for a particular n, say n=k,"},{"Start":"00:45.755 ","End":"00:51.140","Text":"we have to show from this that it must also be true for the next k,"},{"Start":"00:51.140 ","End":"00:52.700","Text":"for k plus 1."},{"Start":"00:52.700 ","End":"00:58.460","Text":"This we show just by substituting n=1 and seeing that we get a true statement."},{"Start":"00:58.460 ","End":"01:00.529","Text":"Sometimes it\u0027s phrased in terms of proposition."},{"Start":"01:00.529 ","End":"01:02.060","Text":"We say that the proposition,"},{"Start":"01:02.060 ","End":"01:05.390","Text":"which in this case is Newton\u0027s binomial theorem for a given n,"},{"Start":"01:05.390 ","End":"01:07.000","Text":"we call it P_n,"},{"Start":"01:07.000 ","End":"01:09.050","Text":"and then there are two things that we have to show,"},{"Start":"01:09.050 ","End":"01:15.140","Text":"step a that P_1 is true, and secondly,"},{"Start":"01:15.140 ","End":"01:20.450","Text":"we show note that P_k is true because that would be going in circles,"},{"Start":"01:20.450 ","End":"01:24.455","Text":"but that if P_k is true, means n=k,"},{"Start":"01:24.455 ","End":"01:29.965","Text":"then it implies that P_k plus 1 is true."},{"Start":"01:29.965 ","End":"01:32.735","Text":"Let\u0027s start with part a,"},{"Start":"01:32.735 ","End":"01:36.515","Text":"which is usually the easy part where n=1,"},{"Start":"01:36.515 ","End":"01:39.373","Text":"and what this says is that,"},{"Start":"01:39.373 ","End":"01:42.005","Text":"well, it\u0027s actually true for n=0."},{"Start":"01:42.005 ","End":"01:44.570","Text":"We are not asked to prove it for n=0 anyway,"},{"Start":"01:44.570 ","End":"01:46.340","Text":"but it is true for n=0."},{"Start":"01:46.340 ","End":"01:47.570","Text":"It just works out."},{"Start":"01:47.570 ","End":"01:51.485","Text":"We only get one term which is equal to 1 and a plus b to the 0 is 1."},{"Start":"01:51.485 ","End":"01:55.145","Text":"We are starting from 1 onwards."},{"Start":"01:55.145 ","End":"02:02.580","Text":"When n=1, what this says is that (a+b)^1 equals,"},{"Start":"02:02.580 ","End":"02:03.900","Text":"this is what we\u0027re checking,"},{"Start":"02:03.900 ","End":"02:05.480","Text":"so I\u0027ll put a question mark here."},{"Start":"02:05.480 ","End":"02:10.100","Text":"The sum k goes from 0 to 1,"},{"Start":"02:10.100 ","End":"02:12.580","Text":"1 choose k,"},{"Start":"02:12.580 ","End":"02:18.155","Text":"a^1 minus k b^k."},{"Start":"02:18.155 ","End":"02:22.400","Text":"Now, this is going to have two terms when k is 0 and K is 1."},{"Start":"02:22.400 ","End":"02:29.430","Text":"When k is 0, we\u0027ll get 1 choose 0, a^1, b^0."},{"Start":"02:29.750 ","End":"02:32.370","Text":"When K is 1,"},{"Start":"02:32.370 ","End":"02:39.900","Text":"then we will get 1 choose 1, a^0, b^1."},{"Start":"02:39.900 ","End":"02:44.220","Text":"What this says is 1 choose 0 is 1."},{"Start":"02:44.220 ","End":"02:45.915","Text":"This is just a^1."},{"Start":"02:45.915 ","End":"02:47.955","Text":"This is just a,"},{"Start":"02:47.955 ","End":"02:53.370","Text":"and this 1 choose 1 is 1 times 1 times b^1."},{"Start":"02:53.370 ","End":"02:55.080","Text":"It\u0027s just b."},{"Start":"02:55.080 ","End":"02:59.025","Text":"This a plus b^1 equal to a plus b?"},{"Start":"02:59.025 ","End":"03:01.490","Text":"The answer is yes."},{"Start":"03:01.490 ","End":"03:03.140","Text":"That\u0027s that part."},{"Start":"03:03.140 ","End":"03:06.455","Text":"Now the more difficult part, the induction step."},{"Start":"03:06.455 ","End":"03:11.915","Text":"Maybe k is also another good letter because we have k here. You know what?"},{"Start":"03:11.915 ","End":"03:19.490","Text":"I\u0027ll change that k to r. I erase the k\u0027s and then I\u0027ll put r\u0027s in their place."},{"Start":"03:19.490 ","End":"03:20.915","Text":"r plus 1."},{"Start":"03:20.915 ","End":"03:24.820","Text":"I don\u0027t want to reuse the same letter for different purposes."},{"Start":"03:24.820 ","End":"03:29.470","Text":"We\u0027re assuming that it\u0027s true for n=r,"},{"Start":"03:29.470 ","End":"03:35.905","Text":"which means that a plus b for a specific r, but general,"},{"Start":"03:35.905 ","End":"03:41.575","Text":"is equal to the sum k goes from 0 to r,"},{"Start":"03:41.575 ","End":"03:43.690","Text":"just replace n by r here."},{"Start":"03:43.690 ","End":"03:51.470","Text":"r choose k a^r minus k and b^k."},{"Start":"03:51.630 ","End":"03:55.840","Text":"We\u0027re given this and by a series of steps,"},{"Start":"03:55.840 ","End":"03:58.430","Text":"we have to show that this is true."},{"Start":"03:58.430 ","End":"04:02.780","Text":"Well, we have to show, I\u0027ll write the endpoint that we\u0027re trying to reach,"},{"Start":"04:02.780 ","End":"04:06.265","Text":"and then by a series of however we can."},{"Start":"04:06.265 ","End":"04:14.270","Text":"To save time, I did a copy-paste and I\u0027m going to replace r by r plus 1,"},{"Start":"04:14.270 ","End":"04:19.760","Text":"r plus 1, r plus 1."},{"Start":"04:19.760 ","End":"04:21.380","Text":"Now the strategy will be,"},{"Start":"04:21.380 ","End":"04:24.680","Text":"we take this statement and we know that it\u0027s true."},{"Start":"04:24.680 ","End":"04:27.515","Text":"That\u0027s the assumption, the induction hypothesis,"},{"Start":"04:27.515 ","End":"04:33.590","Text":"and what we\u0027re going to do is develop the (a+b)^r plus 1,"},{"Start":"04:33.590 ","End":"04:36.275","Text":"which is the left-hand side of what we want to show,"},{"Start":"04:36.275 ","End":"04:38.630","Text":"and by a series of steps,"},{"Start":"04:38.630 ","End":"04:40.745","Text":"I want to reach this expression."},{"Start":"04:40.745 ","End":"04:42.955","Text":"This is like, I don\u0027t know."},{"Start":"04:42.955 ","End":"04:45.920","Text":"I change the color on this because this is what we\u0027re going to"},{"Start":"04:45.920 ","End":"04:49.400","Text":"prove just to emphasize that. Let\u0027s see."},{"Start":"04:49.400 ","End":"04:53.630","Text":"The first thing we can do is something to the power of r plus 1"},{"Start":"04:53.630 ","End":"04:58.510","Text":"using the rules of exponents where 1 plus r is the same as r plus 1."},{"Start":"04:58.510 ","End":"05:06.665","Text":"I can say that this is equal to a plus b times (a+b)^r,"},{"Start":"05:06.665 ","End":"05:09.890","Text":"which is a plus b."},{"Start":"05:09.890 ","End":"05:11.855","Text":"By the induction hypothesis,"},{"Start":"05:11.855 ","End":"05:13.490","Text":"I know what this is."},{"Start":"05:13.490 ","End":"05:14.840","Text":"This is equal to this,"},{"Start":"05:14.840 ","End":"05:20.350","Text":"which I\u0027ll copy-paste here except that we\u0027ll need brackets now."},{"Start":"05:20.350 ","End":"05:22.465","Text":"Use the distributive law,"},{"Start":"05:22.465 ","End":"05:27.950","Text":"will say that a plus b times this is a times this plus b times this."},{"Start":"05:27.950 ","End":"05:31.550","Text":"We have a times this,"},{"Start":"05:31.550 ","End":"05:35.060","Text":"which is here, plus b times."},{"Start":"05:35.060 ","End":"05:37.130","Text":"Next thing I\u0027m going to do is here,"},{"Start":"05:37.130 ","End":"05:38.885","Text":"if I multiply by a,"},{"Start":"05:38.885 ","End":"05:40.580","Text":"which doesn\u0027t depend on k,"},{"Start":"05:40.580 ","End":"05:43.550","Text":"I can multiply this by a,"},{"Start":"05:43.550 ","End":"05:44.975","Text":"put the a in here,"},{"Start":"05:44.975 ","End":"05:48.370","Text":"and so what l have to do is increase this index by 1."},{"Start":"05:48.370 ","End":"05:51.260","Text":"Like I said, we increase the index,"},{"Start":"05:51.260 ","End":"05:53.255","Text":"so I put a plus 1 here."},{"Start":"05:53.255 ","End":"05:54.890","Text":"For the next one,"},{"Start":"05:54.890 ","End":"05:56.260","Text":"I multiply it by b,"},{"Start":"05:56.260 ","End":"05:58.795","Text":"so I increase this index by 1."},{"Start":"05:58.795 ","End":"06:04.010","Text":"Again, I copy-pasted it plus and then I increase this one by 1."},{"Start":"06:04.010 ","End":"06:09.200","Text":"The next step would be to expand the Sigmas to"},{"Start":"06:09.200 ","End":"06:15.505","Text":"write all r plus 1 terms for each of these to main terms."},{"Start":"06:15.505 ","End":"06:18.575","Text":"Some more space. The first part,"},{"Start":"06:18.575 ","End":"06:23.910","Text":"this one will be r choose 0,"},{"Start":"06:24.230 ","End":"06:26.340","Text":"now k is 0 here,"},{"Start":"06:26.340 ","End":"06:28.665","Text":"so it\u0027s just r plus 1,"},{"Start":"06:28.665 ","End":"06:34.635","Text":"and b^0 plus r choose 1,"},{"Start":"06:34.635 ","End":"06:36.780","Text":"then k is 1,"},{"Start":"06:36.780 ","End":"06:43.820","Text":"so we just get a^r b^1."},{"Start":"06:43.820 ","End":"06:45.565","Text":"Then I write third one."},{"Start":"06:45.565 ","End":"06:50.185","Text":"Choose 2 a to the indices here,"},{"Start":"06:50.185 ","End":"06:51.890","Text":"exponents are decreasing by 1,"},{"Start":"06:51.890 ","End":"06:54.740","Text":"so this will be r minus 1,"},{"Start":"06:54.740 ","End":"06:59.490","Text":"b^2, then dot, dot, dot, etc."},{"Start":"06:59.490 ","End":"07:01.335","Text":"I\u0027ll write the last two terms."},{"Start":"07:01.335 ","End":"07:06.220","Text":"We have r minus 1 something and then r,"},{"Start":"07:06.220 ","End":"07:07.890","Text":"r. Let\u0027s see what\u0027s this."},{"Start":"07:07.890 ","End":"07:14.530","Text":"This would be, if k is r minus 1,"},{"Start":"07:14.530 ","End":"07:17.450","Text":"then this computation with k equals minus 1,"},{"Start":"07:17.450 ","End":"07:19.250","Text":"the r will cancel on the 1 will be plus,"},{"Start":"07:19.250 ","End":"07:26.620","Text":"so it\u0027ll be a^2 b^r to the minus 1."},{"Start":"07:26.620 ","End":"07:30.915","Text":"The last one, r choose r,"},{"Start":"07:30.915 ","End":"07:36.075","Text":"will be a^1 b^r."},{"Start":"07:36.075 ","End":"07:39.305","Text":"Now that\u0027s just all this bit."},{"Start":"07:39.305 ","End":"07:45.545","Text":"Plus, now we\u0027ll expand this and I\u0027ll put them in a certain way."},{"Start":"07:45.545 ","End":"07:46.910","Text":"I\u0027ll use a bit of color."},{"Start":"07:46.910 ","End":"07:48.830","Text":"This I\u0027ll write offsets a bit."},{"Start":"07:48.830 ","End":"07:50.720","Text":"I\u0027ll write the first term under this one."},{"Start":"07:50.720 ","End":"07:54.810","Text":"When k is 0, we get r choose 0,"},{"Start":"07:54.810 ","End":"08:00.260","Text":"a^r, and then b^1."},{"Start":"08:00.260 ","End":"08:04.835","Text":"Notice that the coefficients are the same and this is what\u0027s going to happen."},{"Start":"08:04.835 ","End":"08:06.485","Text":"When k is 1,"},{"Start":"08:06.485 ","End":"08:10.380","Text":"then we get r choose 1."},{"Start":"08:10.520 ","End":"08:14.750","Text":"We can just safely just take my word for it. We\u0027ll just quickly do this."},{"Start":"08:14.750 ","End":"08:19.080","Text":"It will come out the same when k is r minus 1."},{"Start":"08:20.840 ","End":"08:26.070","Text":"Here we actually need k to be r minus 2."},{"Start":"08:26.070 ","End":"08:29.480","Text":"You will see because then if this r minus 2,"},{"Start":"08:29.480 ","End":"08:34.445","Text":"then we do indeed get a^2 and k plus 1"},{"Start":"08:34.445 ","End":"08:39.825","Text":"is r minus 2 plus one is r minus 1."},{"Start":"08:39.825 ","End":"08:43.350","Text":"Then we let k be r minus 1,"},{"Start":"08:43.350 ","End":"08:48.180","Text":"so we get r, r minus 1 and here it\u0027s a^1 b^r."},{"Start":"08:48.180 ","End":"08:50.700","Text":"The last term when k is r,"},{"Start":"08:50.700 ","End":"08:52.365","Text":"is r r,"},{"Start":"08:52.365 ","End":"08:57.945","Text":"a^0, b^r plus 1."},{"Start":"08:57.945 ","End":"09:01.950","Text":"Everything aligns except the first and the last."},{"Start":"09:01.950 ","End":"09:05.300","Text":"What we\u0027ll do is to add pairwise."},{"Start":"09:05.300 ","End":"09:12.418","Text":"We\u0027ll add this with this, we\u0027ll add this with this, we\u0027ll add this with this,"},{"Start":"09:12.418 ","End":"09:17.720","Text":"this with this and we have a formula already because we\u0027ve"},{"Start":"09:17.720 ","End":"09:23.990","Text":"already proven previously that we did it with n and k,"},{"Start":"09:23.990 ","End":"09:27.930","Text":"but we could also do it with r and k,"},{"Start":"09:28.990 ","End":"09:32.510","Text":"let\u0027s say this is k minus 1,"},{"Start":"09:32.510 ","End":"09:35.330","Text":"r choose k plus,"},{"Start":"09:35.330 ","End":"09:40.640","Text":"r choose k minus 1 is equal to"},{"Start":"09:40.640 ","End":"09:46.700","Text":"r plus 1 choose k. If we use that formula throughout,"},{"Start":"09:46.700 ","End":"09:49.175","Text":"it\u0027s going to be r plus 1,"},{"Start":"09:49.175 ","End":"09:52.220","Text":"and then it\u0027s going to be the top one of these."},{"Start":"09:52.220 ","End":"09:58.055","Text":"This pair will come out to be r plus 1, choose 1."},{"Start":"09:58.055 ","End":"10:02.395","Text":"They\u0027re all going to start with r plus 1 and we\u0027ll just take the upper one,"},{"Start":"10:02.395 ","End":"10:05.175","Text":"2 r plus 1."},{"Start":"10:05.175 ","End":"10:09.045","Text":"This is r minus 1 here,"},{"Start":"10:09.045 ","End":"10:13.896","Text":"r plus 1, r. Well,"},{"Start":"10:13.896 ","End":"10:14.930","Text":"the first and the last one,"},{"Start":"10:14.930 ","End":"10:16.740","Text":"we can just bring them down,"},{"Start":"10:16.740 ","End":"10:19.885","Text":"r choose 0, and here,"},{"Start":"10:19.885 ","End":"10:23.835","Text":"r, r. I think maybe I\u0027ll change the color."},{"Start":"10:23.835 ","End":"10:26.510","Text":"I think that\u0027s better."},{"Start":"10:26.510 ","End":"10:29.997","Text":"Now, I\u0027ll just copy the a to the something,"},{"Start":"10:29.997 ","End":"10:34.350","Text":"b to the something. Copied that here."},{"Start":"10:34.350 ","End":"10:43.840","Text":"Similarly, a^r b^1 a^r minus 1 b^2, a^2,"},{"Start":"10:43.840 ","End":"10:45.950","Text":"just as common factor,"},{"Start":"10:45.950 ","End":"10:47.350","Text":"I\u0027m just copying it,"},{"Start":"10:47.350 ","End":"10:50.340","Text":"b^r minus 1,"},{"Start":"10:50.950 ","End":"10:57.595","Text":"and a^1,b^r a^0, b^r plus 1."},{"Start":"10:57.595 ","End":"11:01.760","Text":"Now, something\u0027s not quite right because the pattern is broken."},{"Start":"11:01.760 ","End":"11:05.860","Text":"Everything\u0027s r plus 1 except the first and the last,"},{"Start":"11:05.860 ","End":"11:07.895","Text":"but does that really matter?"},{"Start":"11:07.895 ","End":"11:10.925","Text":"The thing is that this is 1,"},{"Start":"11:10.925 ","End":"11:14.090","Text":"and so because it\u0027s 1,"},{"Start":"11:14.090 ","End":"11:21.015","Text":"it wouldn\u0027t hurt if I replace this by r plus 1, like so."},{"Start":"11:21.015 ","End":"11:24.120","Text":"Here r 0 is also 1,"},{"Start":"11:24.120 ","End":"11:28.625","Text":"so I could replace the r by r plus 1, like so."},{"Start":"11:28.625 ","End":"11:32.195","Text":"Now if we write this in Sigma form,"},{"Start":"11:32.195 ","End":"11:36.665","Text":"we can see that this is exactly equal to the sum."},{"Start":"11:36.665 ","End":"11:39.170","Text":"The lower index will be k,"},{"Start":"11:39.170 ","End":"11:43.685","Text":"goes from 0 up to r plus 1."},{"Start":"11:43.685 ","End":"11:50.960","Text":"Here we have a^r plus 1 minus k, r plus 1,"},{"Start":"11:50.960 ","End":"11:53.082","Text":"r plus 1 minus 1 and, so on,"},{"Start":"11:53.082 ","End":"11:56.255","Text":"and the b to the power of k,"},{"Start":"11:56.255 ","End":"12:00.260","Text":"where k goes from 0 up to r plus 1."},{"Start":"12:00.260 ","End":"12:01.670","Text":"So this is right,"},{"Start":"12:01.670 ","End":"12:05.600","Text":"and this is exactly what we were looking for."},{"Start":"12:05.600 ","End":"12:07.310","Text":"Let\u0027s scroll back."},{"Start":"12:07.310 ","End":"12:12.035","Text":"Just take a look at this and let\u0027s scroll back and see what we were looking for."},{"Start":"12:12.035 ","End":"12:13.955","Text":"We were looking for this."},{"Start":"12:13.955 ","End":"12:16.430","Text":"Can I go back down again?"},{"Start":"12:16.430 ","End":"12:18.500","Text":"It doesn\u0027t fit on one screen,"},{"Start":"12:18.500 ","End":"12:21.605","Text":"but this is what we had down below,"},{"Start":"12:21.605 ","End":"12:24.544","Text":"and so this is what we have to prove,"},{"Start":"12:24.544 ","End":"12:26.270","Text":"and we\u0027ve proved it."},{"Start":"12:26.270 ","End":"12:29.910","Text":"That concludes this exercise."}],"Thumbnail":null,"ID":14132},{"Watched":false,"Name":"Exercise 12","Duration":"4m 4s","ChapterTopicVideoID":13492,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.330","Text":"In this exercise, we\u0027re asked to write the seventh term in"},{"Start":"00:04.330 ","End":"00:08.350","Text":"the binomial expansion of this expression."},{"Start":"00:08.350 ","End":"00:12.200","Text":"Of course, we\u0027re going to use Newton\u0027s binomial formula."},{"Start":"00:12.200 ","End":"00:15.460","Text":"Here\u0027s a reminder of the formula."},{"Start":"00:15.460 ","End":"00:18.460","Text":"What we have is that this is a,"},{"Start":"00:18.460 ","End":"00:19.735","Text":"this is b,"},{"Start":"00:19.735 ","End":"00:21.850","Text":"and this is n. Now,"},{"Start":"00:21.850 ","End":"00:24.700","Text":"pay attention the term seventh."},{"Start":"00:24.700 ","End":"00:29.275","Text":"You have to be careful here because the terms start from 0."},{"Start":"00:29.275 ","End":"00:31.765","Text":"This is the kth term."},{"Start":"00:31.765 ","End":"00:33.390","Text":"If we\u0027re going to go to 7,"},{"Start":"00:33.390 ","End":"00:35.280","Text":"we\u0027ll start counting 0, 1,"},{"Start":"00:35.280 ","End":"00:37.740","Text":"2 up to, well,"},{"Start":"00:37.740 ","End":"00:38.990","Text":"if you want 7 terms,"},{"Start":"00:38.990 ","End":"00:41.110","Text":"what I\u0027m saying is it will end up at 6,"},{"Start":"00:41.110 ","End":"00:42.495","Text":"that will be 7 terms."},{"Start":"00:42.495 ","End":"00:47.165","Text":"In other words, we\u0027re going to let n equals 6 in the formula."},{"Start":"00:47.165 ","End":"00:51.560","Text":"Applying it, we just need not the whole formula,"},{"Start":"00:51.560 ","End":"00:54.140","Text":"we just need this term,"},{"Start":"00:54.140 ","End":"01:00.535","Text":"which is this, when k is going to equal 6."},{"Start":"01:00.535 ","End":"01:02.840","Text":"Now what we want is just the term,"},{"Start":"01:02.840 ","End":"01:04.385","Text":"not the whole thing."},{"Start":"01:04.385 ","End":"01:11.375","Text":"We want just this bit when k equals 6."},{"Start":"01:11.375 ","End":"01:14.300","Text":"Now we\u0027ve stated what a and what b,"},{"Start":"01:14.300 ","End":"01:16.040","Text":"and what n are."},{"Start":"01:16.040 ","End":"01:21.250","Text":"All we need, I\u0027ll just copy this term like so,"},{"Start":"01:21.250 ","End":"01:23.525","Text":"it\u0027s wrong color. There we go."},{"Start":"01:23.525 ","End":"01:25.355","Text":"Now it\u0027s just substitutions,"},{"Start":"01:25.355 ","End":"01:29.705","Text":"n is 12, k is 6,"},{"Start":"01:29.705 ","End":"01:31.535","Text":"seventh term, k is 6,"},{"Start":"01:31.535 ","End":"01:36.120","Text":"a is square root of t to the power of,"},{"Start":"01:36.120 ","End":"01:41.610","Text":"n is 12 and k is 6,"},{"Start":"01:41.610 ","End":"01:46.790","Text":"and b is 1 over the square root of"},{"Start":"01:46.790 ","End":"01:52.610","Text":"t. This will take to the power of k, which is 6."},{"Start":"01:52.610 ","End":"01:55.040","Text":"This is what we have to evaluate."},{"Start":"01:55.040 ","End":"01:57.035","Text":"Now the binomial coefficient,"},{"Start":"01:57.035 ","End":"01:58.609","Text":"we\u0027ll do that at the side."},{"Start":"01:58.609 ","End":"02:01.305","Text":"So 12, choose 6,"},{"Start":"02:01.305 ","End":"02:03.050","Text":"so on the denominator,"},{"Start":"02:03.050 ","End":"02:09.545","Text":"we\u0027re going to have 6 times 5 times 4 times 3 times 2 times 1."},{"Start":"02:09.545 ","End":"02:14.805","Text":"Then we start from 12 matching the factors one-on-one, 12,"},{"Start":"02:14.805 ","End":"02:17.310","Text":"11, 10,"},{"Start":"02:17.310 ","End":"02:21.210","Text":"9, 8, 7."},{"Start":"02:21.210 ","End":"02:23.535","Text":"Let\u0027s see what cancels."},{"Start":"02:23.535 ","End":"02:25.620","Text":"6 times 2 is 12,"},{"Start":"02:25.620 ","End":"02:27.300","Text":"so I could do this, this,"},{"Start":"02:27.300 ","End":"02:29.700","Text":"and this. What else?"},{"Start":"02:29.700 ","End":"02:35.525","Text":"5 into 10 goes twice. What else?"},{"Start":"02:35.525 ","End":"02:39.680","Text":"4 into 8 goes twice,"},{"Start":"02:39.680 ","End":"02:45.045","Text":"and 3 into 9 goes 3 times."},{"Start":"02:45.045 ","End":"02:46.785","Text":"There is no denominator."},{"Start":"02:46.785 ","End":"02:48.785","Text":"Now let\u0027s see what we get."},{"Start":"02:48.785 ","End":"02:55.885","Text":"We have 11 times 2 times 3 times 2 times 7."},{"Start":"02:55.885 ","End":"02:58.205","Text":"Maybe I\u0027ll do it on a calculator."},{"Start":"02:58.205 ","End":"03:00.455","Text":"We\u0027ll try doing it without the calculator."},{"Start":"03:00.455 ","End":"03:03.740","Text":"2 times 3 times 2 is 12,"},{"Start":"03:03.740 ","End":"03:05.525","Text":"times 7 is 84."},{"Start":"03:05.525 ","End":"03:08.525","Text":"I\u0027ll just need 11 times 84,"},{"Start":"03:08.525 ","End":"03:14.640","Text":"and that is going to be 840 plus 84."},{"Start":"03:14.640 ","End":"03:19.725","Text":"That would be 924, hopefully."},{"Start":"03:19.725 ","End":"03:24.215","Text":"Back here this we evaluated at 924,"},{"Start":"03:24.215 ","End":"03:25.520","Text":"this is 6,"},{"Start":"03:25.520 ","End":"03:28.055","Text":"and this is also 6."},{"Start":"03:28.055 ","End":"03:30.590","Text":"By chance, this 6 equals this 6,"},{"Start":"03:30.590 ","End":"03:33.725","Text":"so it\u0027s going to come out neater than I planned."},{"Start":"03:33.725 ","End":"03:39.920","Text":"What we get is 924 times square root"},{"Start":"03:39.920 ","End":"03:44.435","Text":"of t^6 times 1 over square root of t^6."},{"Start":"03:44.435 ","End":"03:48.320","Text":"I can take them both to the power of 6."},{"Start":"03:48.320 ","End":"03:51.950","Text":"Now square root of t over square root of t is 1,"},{"Start":"03:51.950 ","End":"03:54.785","Text":"so this thing just comes out to be 1."},{"Start":"03:54.785 ","End":"03:58.220","Text":"We actually get a numerical answer without t,"},{"Start":"03:58.220 ","End":"04:01.675","Text":"which is 924,"},{"Start":"04:01.675 ","End":"04:04.600","Text":"and we are done."}],"Thumbnail":null,"ID":14133},{"Watched":false,"Name":"Exercise 13","Duration":"9m 10s","ChapterTopicVideoID":13493,"CourseChapterTopicPlaylistID":77265,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"This exercise is 1 of the practical applications"},{"Start":"00:04.965 ","End":"00:10.380","Text":"of the Newton\u0027s binomial expansion theorem,"},{"Start":"00:10.380 ","End":"00:13.740","Text":"as a practical well in the age of calculators."},{"Start":"00:13.740 ","End":"00:16.140","Text":"But got to imagine the old days or if you\u0027re"},{"Start":"00:16.140 ","End":"00:18.870","Text":"stuck without a calculator for whatever reason,"},{"Start":"00:18.870 ","End":"00:25.874","Text":"and you want to approximate 0.99^18 to 3 decimal places."},{"Start":"00:25.874 ","End":"00:30.600","Text":"Obviously, we\u0027re going to use Newton\u0027s binomial theorem and here it is."},{"Start":"00:30.600 ","End":"00:32.310","Text":"But how is this?"},{"Start":"00:32.310 ","End":"00:36.270","Text":"This exercise is 1 of the applications of"},{"Start":"00:36.270 ","End":"00:40.320","Text":"Newton\u0027s binomial expansion theorem and"},{"Start":"00:40.320 ","End":"00:44.540","Text":"shows how we can do some approximations without calculator."},{"Start":"00:44.540 ","End":"00:48.170","Text":"This was more useful before the age of the calculator."},{"Start":"00:48.170 ","End":"00:50.810","Text":"We want to compute this to 3 decimal places."},{"Start":"00:50.810 ","End":"00:54.550","Text":"Let\u0027s bring the theorem in. There we go."},{"Start":"00:54.550 ","End":"01:00.949","Text":"Somehow we\u0027re going to choose a and b and n judiciously to help us with this task."},{"Start":"01:00.949 ","End":"01:08.360","Text":"Now the thing is that we do know what is 1^18 and 1 is very close to 0.99."},{"Start":"01:08.360 ","End":"01:15.220","Text":"If we wrote 0.99 as 1 minus 0.01,"},{"Start":"01:15.220 ","End":"01:18.950","Text":"this is 1 minus 0.01,"},{"Start":"01:18.950 ","End":"01:23.750","Text":"then it makes sense to take a=1,"},{"Start":"01:23.750 ","End":"01:28.160","Text":"to take b= minus 0.01,"},{"Start":"01:28.160 ","End":"01:32.935","Text":"and to take n=18."},{"Start":"01:32.935 ","End":"01:35.800","Text":"Then we\u0027ll stop with the expansion and"},{"Start":"01:35.800 ","End":"01:39.380","Text":"normally there\u0027ll be would be a lot of terms, actually 19 terms."},{"Start":"01:39.380 ","End":"01:42.335","Text":"But you\u0027ll see that when we start writing them out,"},{"Start":"01:42.335 ","End":"01:44.330","Text":"they get small very quickly."},{"Start":"01:44.330 ","End":"01:48.170","Text":"Only the first few terms will have any effect on the third decimal place."},{"Start":"01:48.170 ","End":"01:49.685","Text":"Let\u0027s start doing that."},{"Start":"01:49.685 ","End":"01:52.260","Text":"What we have is that 1"},{"Start":"01:52.260 ","End":"02:00.525","Text":"minus 0.01^18 where this is 0.99,"},{"Start":"02:00.525 ","End":"02:03.135","Text":"b is minus 0.01."},{"Start":"02:03.135 ","End":"02:08.690","Text":"This will equal 18 to 0. I don\u0027t know why I write that."},{"Start":"02:08.690 ","End":"02:11.765","Text":"It\u0027s always 1, but just for the formality,"},{"Start":"02:11.765 ","End":"02:16.985","Text":"a is 1^18"},{"Start":"02:16.985 ","End":"02:22.700","Text":"minus 0.01^0."},{"Start":"02:22.700 ","End":"02:26.880","Text":"Next we take 18 on 1,"},{"Start":"02:27.760 ","End":"02:33.810","Text":"1^17 minus 0.01^1,"},{"Start":"02:33.810 ","End":"02:36.840","Text":"it has another couple and if we need more, we\u0027ll add them,"},{"Start":"02:36.840 ","End":"02:40.350","Text":"so 18 choose 2,"},{"Start":"02:40.350 ","End":"02:48.315","Text":"1^16 minus 0.01^2 and 1 more for good measure,"},{"Start":"02:48.315 ","End":"02:56.640","Text":"18 choose 3,1^15,18 minus"},{"Start":"02:56.640 ","End":"03:02.180","Text":"3 and minus 0.01^3."},{"Start":"03:02.180 ","End":"03:05.360","Text":"Now this goes on and we\u0027ll see maybe this will do."},{"Start":"03:05.360 ","End":"03:07.520","Text":"This is 1,"},{"Start":"03:07.520 ","End":"03:08.615","Text":"this is 1,"},{"Start":"03:08.615 ","End":"03:10.115","Text":"this is just 1."},{"Start":"03:10.115 ","End":"03:11.705","Text":"The very first term."},{"Start":"03:11.705 ","End":"03:13.370","Text":"Now the next 1,"},{"Start":"03:13.370 ","End":"03:20.990","Text":"this bit 18 on 1 is 18 and 18 times minus 0.01,"},{"Start":"03:20.990 ","End":"03:23.960","Text":"I\u0027ll just indicate that this computation is 18,"},{"Start":"03:23.960 ","End":"03:29.655","Text":"which we don\u0027t really compute and this is just minus 0.01."},{"Start":"03:29.655 ","End":"03:35.120","Text":"Altogether we get minus 0.18."},{"Start":"03:35.120 ","End":"03:36.950","Text":"The next 1 is obviously going to be a plus,"},{"Start":"03:36.950 ","End":"03:38.090","Text":"it\u0027s a minus squared,"},{"Start":"03:38.090 ","End":"03:42.315","Text":"so I can safely write plus 18 choose 2,"},{"Start":"03:42.315 ","End":"03:48.705","Text":"this bit is 18 times 17 over 2 times 1,"},{"Start":"03:48.705 ","End":"03:54.435","Text":"which is 9 times 17 and that\u0027s 153."},{"Start":"03:54.435 ","End":"03:58.200","Text":"17 times 10 is 170 minus 17."},{"Start":"03:58.200 ","End":"04:04.985","Text":"This thing squared is now 0.01,"},{"Start":"04:04.985 ","End":"04:08.690","Text":"then it\u0027s going to be 0001."},{"Start":"04:08.690 ","End":"04:17.660","Text":"What we get is a 153 times this, which is 0.0153."},{"Start":"04:17.660 ","End":"04:22.960","Text":"But we have to keep going because we\u0027re even in the second decimal place."},{"Start":"04:22.960 ","End":"04:25.400","Text":"Next term is obviously a minus,"},{"Start":"04:25.400 ","End":"04:27.590","Text":"so minus to an odd power."},{"Start":"04:27.590 ","End":"04:36.540","Text":"Now 18 choose 3 over 3 times 2 times 1 and here 18,17,16."},{"Start":"04:36.540 ","End":"04:39.880","Text":"What do we get? 3 times 2 is 6,"},{"Start":"04:39.880 ","End":"04:43.195","Text":"6 into 18 goes 3 times."},{"Start":"04:43.195 ","End":"04:52.610","Text":"It\u0027s 3 times 17 times 16 and that would be 51 times 16."},{"Start":"04:52.610 ","End":"04:56.380","Text":"Well, 50 times 16 is 800,"},{"Start":"04:56.380 ","End":"04:58.810","Text":"so it\u0027ll be 816,"},{"Start":"04:58.810 ","End":"05:04.800","Text":"50 times 16 is 816."},{"Start":"05:04.800 ","End":"05:08.860","Text":"This is already 0.000001."},{"Start":"05:12.650 ","End":"05:16.810","Text":"This will come out, let\u0027s see,"},{"Start":"05:16.810 ","End":"05:24.345","Text":"no need, 000 and 816."},{"Start":"05:24.345 ","End":"05:28.700","Text":"Then it\u0027ll be a plus and it\u0027ll get small very quickly because this thing,"},{"Start":"05:28.700 ","End":"05:32.675","Text":"when the 1 keeps getting further to the right then more and more zeros."},{"Start":"05:32.675 ","End":"05:35.450","Text":"The next term will be a lot less."},{"Start":"05:35.450 ","End":"05:38.090","Text":"It looks like this is in the fourth place."},{"Start":"05:38.090 ","End":"05:39.170","Text":"It could have an influence,"},{"Start":"05:39.170 ","End":"05:41.405","Text":"but the rest of them won\u0027t so we\u0027ll stop here."},{"Start":"05:41.405 ","End":"05:45.010","Text":"What we need to do is evaluate this here."},{"Start":"05:45.010 ","End":"05:46.980","Text":"I\u0027m going to do it without a calculator,"},{"Start":"05:46.980 ","End":"05:54.800","Text":"so 1 minus 0.18 will be 0.82."},{"Start":"05:54.800 ","End":"06:00.635","Text":"That\u0027s this minus this. Then if we add 0.0153,"},{"Start":"06:00.635 ","End":"06:03.860","Text":"we\u0027ll get 0.8,"},{"Start":"06:03.860 ","End":"06:07.530","Text":"and then the 1 with the 2 make it 353."},{"Start":"06:07.530 ","End":"06:11.070","Text":"Then if we subtract this,"},{"Start":"06:11.070 ","End":"06:15.285","Text":"we\u0027re subtracting the 816,"},{"Start":"06:15.285 ","End":"06:20.680","Text":"the 816 comes out 816 here."},{"Start":"06:23.060 ","End":"06:26.585","Text":"I give up, I used the calculator."},{"Start":"06:26.585 ","End":"06:29.495","Text":"Now here I\u0027m in a bit of a dilemma,"},{"Start":"06:29.495 ","End":"06:32.890","Text":"because if I want to round it to 3 places,"},{"Start":"06:32.890 ","End":"06:37.260","Text":"clearly I round down the 4 because this is less than 5,"},{"Start":"06:37.260 ","End":"06:43.050","Text":"but it\u0027s a bit doubtful maybe if the next 1 is a plus."},{"Start":"06:43.050 ","End":"06:46.905","Text":"It might just bring it over the 5 here."},{"Start":"06:46.905 ","End":"06:49.680","Text":"Better do another term."},{"Start":"06:49.680 ","End":"06:54.030","Text":"We\u0027ll add the next term which would be"},{"Start":"06:54.030 ","End":"07:02.585","Text":"18 choose 4 times 1 to the 14th is not going to count,"},{"Start":"07:02.585 ","End":"07:06.553","Text":"and it\u0027s going to be minus 0.01^4,"},{"Start":"07:06.553 ","End":"07:10.325","Text":"so it\u0027s going to be times 0."},{"Start":"07:10.325 ","End":"07:15.915","Text":"That\u0027s going to be what it is to the 4th, 0.01^4."},{"Start":"07:15.915 ","End":"07:18.210","Text":"Well, and the minus, but it\u0027s plus."},{"Start":"07:18.210 ","End":"07:24.940","Text":"This we have to compute 18,17 well that\u0027s was the denominator,"},{"Start":"07:25.880 ","End":"07:33.390","Text":"4,3,2,1,16,15,4 and 2 times is it 8?"},{"Start":"07:33.390 ","End":"07:36.840","Text":"8 into 16 goes twice,"},{"Start":"07:36.840 ","End":"07:43.290","Text":"and 3 into 18 goes 6 times."},{"Start":"07:43.290 ","End":"07:48.825","Text":"6 times 2 is 12,12 times 15 is 180."},{"Start":"07:48.825 ","End":"07:52.645","Text":"I need 180 times 17,"},{"Start":"07:52.645 ","End":"07:57.635","Text":"I make it 3,060,"},{"Start":"07:57.635 ","End":"08:06.840","Text":"so that this bit here will become let\u0027s"},{"Start":"08:06.840 ","End":"08:16.212","Text":"see 3,060 means that this becomes 3,060 so there is 4 zeros and then 306,"},{"Start":"08:16.212 ","End":"08:19.125","Text":"I don\u0027t have to put this 0 at the end."},{"Start":"08:19.125 ","End":"08:27.100","Text":"Now, add these 2 and what we get is 0.8345146."},{"Start":"08:34.100 ","End":"08:37.700","Text":"Now, I wanted to 3 decimal places."},{"Start":"08:37.700 ","End":"08:40.175","Text":"I suppose you could just truncate,"},{"Start":"08:40.175 ","End":"08:42.490","Text":"but that\u0027s not the best thing you can do."},{"Start":"08:42.490 ","End":"08:46.820","Text":"You could say 0.834."},{"Start":"08:46.820 ","End":"08:50.060","Text":"But really, because this is a 5,"},{"Start":"08:50.060 ","End":"08:51.980","Text":"anything that\u0027s 5 and above,"},{"Start":"08:51.980 ","End":"08:56.000","Text":"we should really round this 1 up and not down,"},{"Start":"08:56.000 ","End":"09:02.420","Text":"I think a better answer would be 0.835."},{"Start":"09:02.420 ","End":"09:04.895","Text":"Though you might get away with this."},{"Start":"09:04.895 ","End":"09:10.440","Text":"I\u0027m going to say that this is our answer and we are done."}],"Thumbnail":null,"ID":14134}],"ID":77265}]

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1.1

1