Let $A_n = \{a_1,\dots,a_n\}$ be a sequence of non-decreasing non-negative integers. Let $P(A_n,k)$ be the set of all subsequences of $A_n$ of length $k$. Given $n,k\in\mathbb Z_{\geqslant0}$ with $n\geqslant k$ I would like to find, in general,
all the possible values of $\sum_{j=1}^k b_{i,j}$ for every $B_i=\{b_{i,1},\dots,b_{i,k}\}\in P(A_n,k)$, and
how many elements of $P(A_n,k)$ attain each specific sum value.
I know that $P(A_n,k)$ has size $\binom nk$, but beyond that, I don't know what tools to use (combinatorics is not my main field) to get the answers. Also the fact that numbers can repeat seems to make it more difficult. I tried running some small examples, but I couldn't discern any pattern.