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Let us suppose to have a set of N elements and to choose m elements out of the set. I know I can do it in $ N \choose m $ ways; but I want to know how many of these combinations contain k predefined elements (say 1,2,...,k).

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up vote 7 down vote accepted

If you fix k elements, you are left with m-k elements to choose among N-k elements, which can be done in $N-k \choose m-k$ ways.

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