Vandermonde's identity

In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients:

${\displaystyle {m+n \choose r}=\sum _{k=0}^{r}{m \choose k}{n \choose r-k}}$

for any nonnegative integers r, m, n. The identity is named after Alexandre-Théophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie.^[1]

There is a q-analog to this theorem called the q-Vandermonde identity.

Vandermonde's identity can be generalized in numerous ways, including to the identity

${\displaystyle {n_{1}+\dots +n_{p} \choose m}=\sum _{k_{1}+\cdots +k_{p}=m}{n_{1} \choose k_{1}}{n_{2} \choose k_{2}}\cdots {n_{p} \choose k_{p}}.}$