Probability of having k defined elements while choosing m elements from a set of N

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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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.