Give a story proof that
$\sum_{k=0}^n k{n\choose k}^2 = {n{2n-1\choose n-1}}$
Consider choosing a committee of size n from two groups of size n each , where only one of the two groups has people eligible to become president.
So I think the left side represents the number of ways to choose groups when you have k = 1,2,3…people chosen from the "president-eligible" group, and the rest from the other group, and then sums up the possibilities. On the right hand side, n people are eligible to become president, and then you choose the rest of the group members? Or something like that? What is the best interpretation? Thanks.
What is the connection with Vandermonde's identity? (this was suggested on another post with a similar proof)
Best Answer
Hint:
$$\sum_{k=0}^n k{n \choose k}^2 = \sum_{k=0}^n \underbrace{ k \ \ \ {n \choose k}}_{\text{Pres eligible group}}{n \choose n - k}$$