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Given a set $S=\{1..n\}$ and all it's subsets $Q$ of size $m$, we have:

$|Q| = \binom{n}{m}$

I would like to find a generic formula for the sum of products of all elements in each of the subsets. E.g. for $n=4, m= 2$:

$1 * 2 + 1 * 3 + 1 * 4 + 2 * 3 + 2 * 4 + 3 * 4$

Denis Kulagin
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1 Answers1

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These are Stirling numbers of the first kind. These satisfy the generating function $$x(x+1)(x+2)\cdots(x+n-1)=\sum_{k=1}^n s(n,k)x^k.$$ So the solution to your problem is $s(n+1,n+1-k)$.

Angina Seng
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