Let x+y=k where x, y>0 and S(k, n)=∑r=0^n r^2( ^n Cr) x^' y^n-r then (a) S(1,5

step 1 Here, we're given this series here, S -Q2 and equal this. We're to find out S -15, 2, 3, and 44.

Let x+y=k where x, y>0 and S(k, n)=∑r=0^n...

Key Concepts

Binomial Theorem

The binomial theorem provides the expansion of (x + y)^n as a sum of terms of the form (n choose r)x^r y^(n-r). This theorem is essential in linking combinatorial coefficients with algebraic expressions and is the foundation for many identities involving sums over binomial coefficients.

Differentiation of Generating Functions

By differentiating the binomial expansion with respect to x (or y), one can derive useful identities involving sums that include factors of r. For example, the first derivative yields an expression for the sum of r (n choose r)x^(r-1) y^(n-r), and a second differentiation yields a related expression involving r(r-1). These techniques are vital for evaluating sums weighted by powers of the summation index.

Combinatorial Identities for r-Power Sums

Often in problems involving sums of the form ? r^2 (n choose r)x^r y^(n-r), it is useful to rewrite r^2 as r(r-1) + r. This allows one to use the identities obtained from the first and second derivatives of the binomial expansion. Such identities are frequently used in combinatorics and probability, notably in computing moments of the binomial distribution.

Transcript

00:02 Here, we're given this series here, s -q2 and equal this.

00:06 We're to find out s -15, 2, 3, and 44.

00:10 And we're given here x plus y equals k, x, y, greater than 0.

00:16 Now, for that, so here we have it is 1, 5.

00:20 So we start with here with k equal to 1.

00:24 Before that, you simplify this, r square and the r.

00:30 Then we simplify this, then you put into this summation, and then find out the summation here.

00:35 So here we have it is r square basically and here we have ncr.

00:39 So first this need to be simplified.

00:41 Otherwise, we're not able to find out of summation here.

00:44 Okay, equal to 1.

00:45 So let's work on r square ncr.

00:56 So r square can be given as r, r, r negative 1 plus r times being as nc.

01:07 This is you got now.

01:08 So this can be now distributed as so we get here r negative 1 times ncr.

01:17 That is n factorial over a negative r factorial times r factorial plus we have r times n c r again we can write n factorial over n negative r factorial times r factorial now let's work on this so this is r factorial here we have this is r r negative 1 if we do just look at some cancellations here it will be r r negative 1 and here we get r negative 2 factorial so write this as n factorial over we have n negative r factorial times r negative two factorial now this n negative r here this we can write it as n negative 2 negative so let's write that here and also we can write n factorial as n n negative 1 times n negative 2 factorial.

02:28 Let's first work on this.

02:29 So we have here n, n negative 1 and negative 2, factorial over we have n negative 2 negative r negative 2, all factorial times r negative 2 factorial.

02:44 Now this part here, n negative 2 factorial over and negative r negative 2 factorial times r negative to vector this will get n negative 2 c r negative 2 so we get here n times n negative 1 then we have n negative 2 c r negative 2 similarly we're to simplify this so this we get at so here we have r factorial so we get here it is plus we get n times n negative 1 factorial over we have here r negative 1 factorial then this n negative r this can written as n negative 1 negative r negative r negative r negative 1 factorial so to see now this n negative 1 factorial here over n negative negative r negative 1 factorial times r negative 1 factorial that can be given as n negative 1 c r negative 1 so we get here n times n negative 1 p n negative 1 p b r negative 1.

04:09 Now we substitute this value for r square ncr in the given expression for the firm series for k equal to 1.

04:19 So we get s 1 n that equals n times n negative 1 we have summation r equal to to n negative 2 c r negative 2 x to power r y to the power n negative r next we get plus and summation r equal 1 to n n negative 1 c r negative 1 x to bar r r to bar r the question is why we have changed this here from r equal to 0 it is given us here are equal to 0 and the change is 2 here r equal 2 and there are equal to 1 why we have changed so we have because here we have it is r negative 2 so the first term is starting from here equal and here starting from r equals 1.

05:19 Therefore, starting from r equal to 1, and here starting from r equal to 2.

05:24 Otherwise, if it would r equal to 1 or 0, it will become negative.

05:28 So then combination will not be valid.

05:32 Now, if it's observed this here, here we have x to power r.

05:37 Here we have i to the bar n negative r.

05:40 What you can do here in the next step, we'll write n, n, n negative 1, summation, r equals 2 to n negative 2 c r negative 2 we do here extra power r negative 2 times x square and times y to the power and negative r now to see here this series is coming out to be y plus x or also we can write it just here we have that can be given as n negative 2 negative r negative r negative 2 this part.

06:25 So this series here, this theory, this can be given as y plus x, the power we have n negative 2.

06:40 Why? because it's just open it out for the r -f term that is given as n negative 2.

06:46 We have the r -negative 2...

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