Binomial (Properties of Coefficients)

[ritwik06]

Homework Statement

$$\sum^{n}_{r=0} (2r+1) (^{n} C_{r})^{2}$$

The Attempt at a Solution

$$x(1+x^{2})^{n}$$
If I differentiate this and put x=1;
I will get the above series without the squares of the binomial coefficients.Will multiplying by $$(1+x)^{n}$$ help now?

 

[HallsofIvy]

The "problem" you give is a polynomial. Now, what are you supposed to do with it? What is the question?

 

[ritwik06]

I have to prove this equal to anyone of these.
$$a) (2n+2) ^{2n} C_{n}$$

$$b) (n+1) ^{2n} C_{n}$$

$$c) (2n+1) ^{2n} C_{n}$$

$$d) (n) ^{2n} C_{n}$$

 

[ritwik06]

Help me!

 

[Defennder]

Well to make things easier (and to cheat a little), let n=2, for example. You should find that only (b) holds. Now of course a proof is required, so that itself doesn't count. But at least you know where to focus your effort.

 

[ritwik06]

Well to make things easier (and to cheat a little), let n=2, for example. You should find that only (b) holds. Now of course a proof is required, so that itself doesn't count. But at least you know where to focus your effort.

The answer was already given to me in the textbook. I am just wondering how to prove the result...

 

[Dick]

Roughly the same trick as the other one. Replace r in 2r+1 by n-r. As for C(n,r)^2, That's the same as C(n,r)*C(n,n-r). If you sum of over r, isn't that the same as the number of ways to choose n objects from a group of 2n objects?