Vandermonde's Identity - Combinatorial Proof

Combinatorial Proof

Vandermonde's identity also admits a more combinatorics-flavored double counting proof, as follows. Suppose a committee in the US Senate consists of m Democrats and n Republicans. 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 Democrats and rk Republicans:

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