# Binomial (Properties of Coefficients)

## 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?

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

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

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

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

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

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Help me!!

Defennder
Homework Helper
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.

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 text book. I am just wondering how to prove the result....

Dick