# The number of sequences with k elements, containing a given element

Given a sequence ${a_1, a_2, ..., a_n}$ and a number k I have to find the number of subsequences with k elements, containing the element $a_k$. I know that the total number of all subsequences is $2^n$, but it doesn't seem to be very helpful here.

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## 1 Answer

If you are given that each sequence contains $a_k$ and you have $k$ elements total in the sequence, you only need $k-1$ other elements to fill up the rest of your sequence with, and you have $n-1$ other elements to choose from. It should be easier to count when you think about it like this.

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Is the answer just $C^{k-1}_{n-1}?$ –  Ormi Nov 24 '12 at 23:06
@UchihaMadara exactly! –  Tom Oldfield Nov 25 '12 at 0:22