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Michael
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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.

Michael
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