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Vandermonde's identity
Mathematical theorem on convolved binomial coefficients
Vandermonde's identity
In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: (m+nr)=∑k=0r(mk)(nr−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 (n1+⋯+npm)=∑k1+⋯+kp=m(n1k1)(n2k2)⋯(npkp).
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