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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:
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
In general, the product of two polynomials with degrees m and n, respectively, is given by
where we use the convention that ai = 0 for all integers i > m and bj = 0 for all integers j > n. By the binomial theorem,
Using the binomial theorem also for the exponents m and n, and then the above formula for the product of polynomials, we obtain
where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all i > m and j > n, respectively.
By comparing coefficients of x r, Vandermonde's identity follows for all integers r with 0 ≤ r ≤ m + n. For larger integers r, both sides of Vandermonde's identity are zero due to the definition of binomial coefficients.
Vandermonde's identity also admits a combinatorial double counting proof, as follows. Suppose a committee consists of m men and n women. In how many ways can a subcommittee of r members be formed? The answer is
The answer is also the sum over all possible values of k, of the number of subcommittees consisting of k men and r − k women:
Take a rectangular grid of r x (m+n−r) squares. There are
paths that start on the bottom left vertex and, moving only upwards or rightwards, end at the top right vertex (this is because r right moves and m+n-r up moves must be made (or vice versa) in any order, and the total path length is m + n). Call the bottom left vertex (0, 0).
There are paths starting at (0, 0) that end at (k, m−k), as k right moves and m−k upward moves must be made (and the path length is m).
Official source
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