I stumbled across this identity involving binomial coefficients this morning:

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

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

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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    $\begingroup$ So-called "alternate" proof: equate coeffients in the identity $(1+x)^a (1+x)^b = (1+x)^{a+b}$. $\endgroup$ – GEdgar Sep 29 '11 at 19:35
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    $\begingroup$ 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. $\endgroup$ – J. M. is a poor mathematician Sep 29 '11 at 19:49

That's Vandermonde's identity.


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