Combinatorial proof of $\sum \limits_{k=1}^n k {n\choose k}^2 = n {2n-1 \choose n-1}.$ [closed]

Question

Give a combinatorial proof & proof using generating functions of the following identity:

$$\sum_{k=1}^n k {n\choose k}^2 = n {2n-1 \choose n-1}.$$

Do you know what the terms mean? Can you give us some background? Do you know the corresponding proofs of the Vandermonde identity?
Well, if you don't know whether you know what $C(n,k)$ means or what the summation index of your sum is, or whether you know the corresponding proof of the Vandermonde identity, I don't know why you are asking the question. Do you think that I will explain your question to you?
As you alluded, it looks like the Vandermonde identity. Look at the proofs (combinatorial and generating function respectively) of that identity and see how you can deal with the $k$.
Please do not post questions in the imperative mode. You are not assigning us homework. Rather than "give a proof..." say "how do I prove...."

Phira · Answer 1 · 2011-04-23 21:28:00Z

up vote 5 down vote

Combinatorial proof:

Adapt the combinatorial proof for the Vandermonde identity by kissing one of the chosen men.

Proof by generating functions:

Adapt the proof for the Vandermonde identity by differentiating both sides.

answered Apr 23 '11 at 21:28

Phira
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2

Kissing? Men? Not president of the committee? – Mitch Apr 24 '11 at 1:25

closed as not a real question by Andres Caicedo, t.b., Eric, Zev Chonoles, J. M. Apr 26 '11 at 12:54