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I stumbled across this identity involving binomial coefficients this morning:

If $n$, $k$, $a$, and $b$ are positive integers and $n=a+b$, then

$ \displaystyle \binom{n}{k} =\sum_{i=0}^k \binom{a}{k-i}\binom{b}{i}$.

The proof is trivial. (Partition a set $N$ of $n$ things into disjoint sets $A$ of $a$ things and $B$ of $b$ things. Now to pick $k$ items from $N$, either all $k$ are from $A$, or $k-1$ are from $A$ and $1$ is from $B$, or...)

Has this identity appeared before (I expect the answer to be "Yes!") and if so, does it have a name?

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So-called "alternate" proof: equate coeffients in the identity $(1+x)^a (1+x)^b = (1+x)^{a+b}$. –  GEdgar Sep 29 '11 at 19:35
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In other words: show that the coefficients of the product of two generating functions is the convolution of the coefficients of the two generating functions. –  J. M. Sep 29 '11 at 19:49

1 Answer 1

up vote 10 down vote accepted

That's Vandermonde's identity.

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Chu-Vandermonde, if we must be complete... ;) –  J. M. Sep 29 '11 at 19:20
    
Great. Thanks!! –  wckronholm Sep 29 '11 at 20:17

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