Let $S=\{1,2,3,...,n\}$ be the set of integers up to $n$ and $p\_k(a\_1,...,a\_k)=a\_1\cdots a\_k$ the product of $k$ distinct integers $a\_1,...,a\_k \in S$. There are $\binom{n}{k}$ possibilities to construct such a product $p\_k$. I was wondering if it is anyhow possible to estimate the sum of all such $k$-products $p\_k$ or similarly the mean value as $n$ is large. Any idea?
Thanks



Sum of all products of k distinct integers in [1,n]
The sum of all $k$-products of numbers in the interval $\{1,2,\ldots,n\}$, it is, the number you are referring to, is known as the Stirling number of the first kind: ${n+1 \brack {n+1-k}}$. There are plenty of articles that face the problem of giving estimations of these numbers out there.


