# Chu-Vandermonde Identity/Proof 1

## Theorem

$\ds \sum_{k \mathop = 0}^n \binom r k \binom s {n - k} = \binom {r + s} n$

## Proof

 $\ds \sum_{n \mathop = 0}^{r + s} \binom {r + s} n x^n$ $=$ $\ds \paren {1 + x}^{r + s}$ Binomial Theorem - Integral Index $\ds$ $=$ $\ds \paren {1 + x}^r \paren {1 + x}^s$ Exponent Combination Laws $\ds$ $=$ $\ds \sum_{k \mathop = 0}^r \binom r k x^k \sum_{k \mathop = 0}^s \binom s k x^k$ Binomial Theorem - Integral Index $\ds$ $=$ $\ds \sum_{n \mathop = 0}^{r + s} \paren {\sum_{k \mathop = 0}^n \binom r k \binom s {n - k} } x^n$ Product of Absolutely Convergent Series

Therefore:

$\ds \binom {r + s} n = \sum_{k \mathop = 0}^n \binom r k \binom s {n - k}$

$\blacksquare$

## Source of Name

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