Chu-Vandermonde Identity

Theorem
Let $r, s \in \R, n \in \Z$.

Then:
 * $\displaystyle \sum_k \binom r k \binom s {n-k} = \binom {r+s} n$

where $\displaystyle \binom r k$ is a binomial coefficient.

Proof 1
As this has to be true for all $x$, we have that:
 * $\displaystyle \binom {r+s} n = \sum_k \binom r k \binom s {n-k}$

Proof 2
Special case of Gauss's Hypergeometric Theorem:


 * $\displaystyle {}_2F_1(a,b;c;1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$

$\displaystyle {}_2F_1$ is the Hypergeometric Series and $\Gamma(n+1)=n!$ is the Gamma function.

One regains the Chu-Vandermonde identity by taking $a = -n$ and applying the identity


 * $\displaystyle \binom n k = (-1)^k \binom {k-n-1} k$

liberally.

Also known as
When $r$ and $s$ are integers, it is more commonly known as Vandermonde's Identity or Vandermonde's Convolution.

Comment
This can be interpreted as follows.

The RHS can be thought of as the number of ways to select $n$ people from among $r$ men and $s$ women.

Each term in the LHS is the number of ways to choose $k$ of the men and $n - k$ of the women.

It appeared in Chu Shih-Chieh's The Precious Mirror of the Four Elements in 1303.

It was published by Vandermonde in 1772.