1 00:00:00,760 --> 00:00:04,710 Hello and welcome to the second lecture on the binomial theorem. 2 00:00:04,710 --> 00:00:11,710 In this lecture, we will look at the binomial theorem and some of the related results which 3 00:00:11,710 --> 00:00:13,830 can be derived from the binomial theorem. 4 00:00:13,830 --> 00:00:23,480 First, recall the definition of nC_r, which is the number of ways of choosing r objects 5 00:00:23,480 --> 00:00:31,660 from n distinct objects, and which can be written as n!/[(n-r)! 6 00:00:31,660 --> 00:00:36,640 * r!]. 7 00:00:36,640 --> 00:00:37,640 So the binomial theorem 8 00:00:40,640 --> 00:00:52,139 given by this result: 9 00:00:52,139 --> 00:01:26,730 (x+y)^n = nC_0 x^n y^0 + nC_1 x^{n-1} y^1 + ... + nC_{n-1} x^1 y^{n-1} + nC_n x^0 y^n, 10 00:01:26,730 --> 00:01:36,320 where nC_r is the number of ways of choosing r objects from n distinct objects. This can 11 00:01:36,320 --> 00:01:50,870 be more succinctly written as the summation \sum_{r=0}^{n}{nC_r x^{n-r} y^r}. 12 00:01:50,870 --> 00:01:57,430 The binomial theorem can be taken to be an identity in x and y. The binomial theorem 13 00:01:57,430 --> 00:02:13,569 is applicable for x and y complex, and n being a positive integer. 14 00:02:13,569 --> 00:02:22,840 At this point, I would like to mention that generalizations of the binomial theorem for 15 00:02:22,840 --> 00:02:33,340 the case when n is any real number also exist. In the definition/in the expression of the 16 00:02:33,340 --> 00:02:45,510 binomial theorem, we take x^0 to be equal to 1 for all x which are complex numbers, 17 00:02:45,510 --> 00:02:53,480 i.e., irrespective of the value of x, we define x^0 to be equal to 1. 18 00:02:53,480 --> 00:03:05,860 Notice that there are n+1 terms in the binomial theorem, and there are NOT n terms but n+1 19 00:03:05,860 --> 00:03:11,790 terms in the binomial theorem. You notice that there is a sort of pattern to the terms 20 00:03:11,790 --> 00:03:18,189 in the binomial theorem, and that is well captured by the general term of the binomial 21 00:03:18,189 --> 00:03:26,680 theorem. In general, the (r+1)th term in the expansion of (x+y)^n can be written as nC_r 22 00:03:26,680 --> 00:03:37,120 x^{n-r} y^r. To 23 00:03:37,120 --> 00:03:47,439 give you an example of the application of the binomial theorem, let's look at (x+y)^1. 24 00:03:47,439 --> 00:03:59,739 From the binomial theorem, we get that this is nothing but 1C_0 x^1 y^0 + 1C_1 x^0 y^1 25 00:03:59,739 --> 00:04:13,040 which gives us x+y, which is what we expect. We can also derive for the case when we have 26 00:04:13,040 --> 00:04:34,620 (x+y)^2, which is nothing but 2C_0 x^2 y^0 + 2C_1 x^1 y^1 + 2C_2 x^0 y^2. Since 2C_0 27 00:04:34,620 --> 00:04:46,750 is 1, we have x^2; 2C_1 is nothing but 2, we have 2xy; and 2C_2 is again 1 so we have 28 00:04:46,750 --> 00:04:55,480 y^2. This is the well known result (x+y)^2 = x^2 + 2xy + y^2. 29 00:04:55,480 --> 00:05:14,420 Similarly, you can derive that (x+y)^3 is x^3 + 3x^2 y + 3x y^2 + y^3, and you can derive 30 00:05:14,420 --> 00:05:19,770 this result from the binomial expansion as well. 31 00:05:19,770 --> 00:05:27,170 So one way in which you can interpret the binomial theorem is as follows: to derive 32 00:05:27,170 --> 00:05:43,750 (x+y)^n is equal to this expression, just consider (x+y)^n as nothing but (x+y)*(x+y)*...*(x+y) 33 00:05:43,750 --> 00:05:58,330 for a total of n terms, and the interpretation of this expansion can be given as follows. 34 00:05:58,330 --> 00:06:04,960 If you choose n x's, then you choose an x from each of these, uh, each of these parentheses 35 00:06:04,960 --> 00:06:10,660 here, and so there is only one way of choosing x's from each of these parentheses. There 36 00:06:10,660 --> 00:06:18,390 is only one way of choosing n x's and 0 y's, and that gives you the coefficient of x^n 37 00:06:18,390 --> 00:06:25,860 y^0. If you want to choose (n-1) x's from this expression, then you have to choose (n-1) 38 00:06:25,860 --> 00:06:33,620 x's from the n possible x's, and you have to choose 1 y from the n possible y's, and 39 00:06:33,620 --> 00:06:41,580 that can be done in n choose 1 ways, and that gives you the coefficient of x^{n-1} y^1 (and 40 00:06:41,580 --> 00:06:46,290 so on). In this way, you can derive the/you can provide 41 00:06:46,290 --> 00:06:51,110 a combinatorial interpretation of the binomial expansion. 42 00:06:51,110 --> 00:07:00,580 So, as I mentioned previously, there are generalizations for the binomial theorem for the case when 43 00:07:00,580 --> 00:07:12,450 this exponent n is not necessarily a positive integer, but when it can be any real number. 44 00:07:12,450 --> 00:07:17,470 This is the general result of the binomial theorem, and you can derive several related 45 00:07:17,470 --> 00:07:23,430 results by playing around with the general result of the binomial theorem. For instance, 46 00:07:23,430 --> 00:07:35,060 you could replace y by -y in the theorem and, upon doing so, you would get an expression 47 00:07:35,060 --> 00:07:54,090 for (x-y)^n. This is going to be nC_0 x^n (-y)^0 + nC_1 x^{n-1} (-y)^1 + ... + nC_n 48 00:07:54,090 --> 00:08:06,620 x^0 (-y)^n, and this can be more simply written as summation 49 00:08:06,620 --> 00:08:22,360 \sum_{r=0}^{n}{(-1)^r nC_r x^{n-r} y^r}. This result is very similar to the result we derived 50 00:08:22,360 --> 00:08:28,139 for the binomial theorem. A second result which we can derive is by 51 00:08:28,139 --> 00:08:49,940 replacing y by 1, in which case we can derive an expansion for (x+1)^n as nC_0 x^n 1^0 + 52 00:08:49,940 --> 00:09:08,960 nC_1 x^{n-1} 1^1 + ... + nC_n x^0 1^n. This can be written as nC_0 x^n + nC_1 x^{n-1} 53 00:09:08,960 --> 00:09:24,060 + ... + nC_n x^0. Now, to arrive at a more simpler expression of this result, use the 54 00:09:24,060 --> 00:09:32,130 fact that nC_r is nC_{n-r} and you can derive this to be equal to nC_n x^n + nC_{n-1} x^{n-1} 55 00:09:32,130 --> 00:09:44,740 + ... + nC_0 x^0. This last expression can simply be written as summation 56 00:09:44,740 --> 00:09:55,860 \sum_{r=0}^{n}{nC_r x^r}. This is the binomial expansion of (x+1)^n. 57 00:09:55,860 --> 00:10:06,920 Similarly, we can also derive this result from the binomial expansion: we can derive 58 00:10:06,920 --> 00:10:17,490 an expansion for (x+y)^n + (x-y)^n. If you apply the binomial expansion for these two 59 00:10:17,490 --> 00:10:39,550 separate expressions, you will get 2*[nC_0 x^n y^0 + nC_2 x^{n-2} y^2 + ...]. The 60 00:10:39,550 --> 00:10:44,990 alternate terms will cancel out and it will leave you with this result. 61 00:10:44,990 --> 00:10:52,370 So that's it for this lecture. Hope you had fun listening to the binomial theorem and 62 00:10:52,370 --> 00:10:58,051 some of its related results. In the next lecture, we will be looking at some examples of the 63 00:10:58,051 --> 00:11:03,240 usage of the binomial theorem and some possible problems you will face in an exam situation. 64 00:11:03,240 --> 00:11:03,890 Thank you.