Name of binomial identity

Asked Jun 15 '22 at 11:36

Active Jun 15 '22 at 14:13

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Does anyone recall the name of the following binomial identity?

Let $k_1,\ldots,k_m$ be non-negative integers and $n\geq \sum_{i=1}^mk_i$. Then

$$\binom{n+m-1}{k_1+k_2+\ldots+k_m+m-1}=\sum_{n_1+n_2+\ldots+n_m=n}\binom{n_1}{k_1}\cdot\ldots\cdot\binom{n_m}{k_m}$$

I used that for some computation regarding the negative hypergeometric distribution. It describes a combinatorial way to create a committee with at least $k_1,\ldots,k_m$ members from $m$ groups.

edited Jun 15 '22 at 14:13

asked Jun 15 '22 at 11:36

Peter Strouvelle

1

It looks like a sort of upside-down Vandermonde's identity to me, but I don't know of a name for it. – Jakob Streipel Jun 15 '22 at 15:04
1

In (unsuccessfully) looking for a name I found some friends: this and this (in which Brian M. Scott also calls it an upside-down Vandermonde identity... maybe it'll catch on!). – Jakob Streipel Jun 15 '22 at 15:11

Name of binomial identity

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