Chu-Vandermonde Identity

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

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

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

Alternative Proof
Special case of Gauss's Hypergeometric Theorem:


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

$$\;_2F_1$$ 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


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

liberally.

Comment
This can be interpreted as follows.

The RHS is 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.