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