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mathematical analysis > combinatorics > Vandermonde's identity

algebra > combinatorics > Vandermonde's identity

algebra > elementary algebra > identity > Vandermonde's identity

Preferred term

Vandermonde's identity

Definition(s)

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.
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

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

Broader concept(s)

Synonym(s)

Vandermonde's convolution

In other languages

identité de Vandermonde

French
formule de convolution

URI

http://data.loterre.fr/ark:/67375/PSR-PJ28VRBP-W

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