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Concept# Vandermonde's identity

Summary

In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients:
:{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
:
{ 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}.
Proofs
Algebraic proof
In general, the product of two polynomials with degrees m and n, respectively, is given by
:\biggl(\sum_{i=0}^m a_ix^i\biggr) \biggl(\sum_{j=0}^n b_jx^j\biggr)
= \sum_{r=0}^{m+n}\biggl(\sum_{k=0}^r a_k b_{r-k}\biggr) x^r,
where we use the convention that ai&nb

Official source

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