Math 2.0

Combinatorics Basic counting Theorem

Vandermonde's identity

Builds on Binomial coefficient

Statement

Let mm, nn and rr be natural numbers. Then (m+nr)=∑k=0r(mk)(nr−k).\binom{m + n}{r} = \sum_{k=0}^{r} \binom{m}{k} \binom{n}{r - k}.

Remarks

Combinatorial proof: take a set of mm men and nn women. An rr-element committee has some number kk of men and r−kr - k women. With m=n=rm = n = r, and the symmetry of binomial coefficients, it gives (2nn)=∑k=0n(nk)2\binom{2n}{n} = \sum_{k=0}^{n} \binom{n}{k}^2.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Vandermonde's_identity, lead section
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Kindstatement, theorem
WritingW1 stated
MSC05A19
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