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Suppose we have list of $n$ integers $i_1,i_2,\ldots, i_n$ , and we choose $k$ integers from it in all possible combinations (i.e., $C^n_k$ combinations). Now for each combination, we multiply chosen $k$ integers, and add the results.

How do I mathematically formulate this? (some form like $ \sum \prod ??? $)

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1 Answer 1

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Let $I=\{i_1,\ldots,i_n\}$, and let $P_k(I)$ be the set of $k$-element subsets of $I$. Then you're describing $$\sum_{S\in P_k(I)}\prod S$$

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Is $P_k(I)$ standard notation? Or I need to specify the notation definition when writing in some journal. –  anuj919 Jan 4 '13 at 15:24
    
@anuj919: You should specify. Another notation often found (and in fact more common in the mathematics that I read) is $[I]^k$. In general, if $S$ is any set, and $\kappa$ is any cardinal number, finite or infinite, $[S]^\kappa$ is the set of all subsets of $S$ of cardinality $\kappa$. You’ll also see $[S]^{<\kappa}$ and $[S]^{\le\kappa}$, with the obvious meanings. –  Brian M. Scott Jan 4 '13 at 15:29
    
Thanks for clarification @brian. –  anuj919 Jan 4 '13 at 15:32
    
@anuj919: My pleasure. –  Brian M. Scott Jan 4 '13 at 15:32

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