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.

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

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

Alternative Proof
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.

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.