For the sums the answer should be approximately $C^k_n/m$ (rounded down, to be exact), where $C^k_n=\frac{n!}{k!(n-k)!}$ is the total number of $S_i$s. 

For a similar question for the products $\Pi(S_i)=a_{i1}...a_{ik}$ the answer would be more interesting.