# 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

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle \sum_n \binom {r + s} n x^n$$ $$=$$ $$\displaystyle$$ $$\displaystyle \left({1 + x}\right)^{r + s}$$ $$\displaystyle$$ $$\displaystyle$$ Binomial Theorem $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \left({1 + x}\right)^r \left({1 + x}\right)^s$$ $$\displaystyle$$ $$\displaystyle$$ Exponent Combination Laws $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \sum_k \binom r k x^k \sum_m \binom s m x^m$$ $$\displaystyle$$ $$\displaystyle$$ Binomial Theorem $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \sum_k \binom r k x^k \sum_{n-k} \binom s {n - k} x^{n - k}$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \sum_n \left({\sum_k \binom r k \binom s {n-k} }\right) x^n$$ $$\displaystyle$$ $$\displaystyle$$

As this has to be true for all $x$, we have that:

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

$\blacksquare$

## Proof 2

Special case of Gauss's Hypergeometric Theorem:

${}_2F_1 \left({a, b; c; 1}\right) = \dfrac{\Gamma \left({c}\right) \Gamma \left({c - a - b}\right)} {\Gamma \left({c - a}\right) \Gamma \left({c - b}\right)}$

where:

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

One regains the Chu-Vandermonde Identity by taking $a = -n$ and applying Negated Upper Index of Binomial Coefficient:

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

throughout.

$\blacksquare$

## Also known as

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

## Notes

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.

## Source of Name

This entry was named for Alexandre-Théophile Vandermonde and Chu Shih-Chieh.

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