Frequency of Item in Combination

Question

Given a set of $m$ characters, if I pick $n$ of them, how many combinations contain a given character?

A simple case: Picking two letters from ABCD has $\binom{4}{2} = 6$ possibilities, AB, AC, AD, BC, BD, CD and each letter appears 3 times. How does one solve for 3, the frequency of an individual letter, given a general $m$ and $n$?

Answer 1

Since it looks like you're sampling without replacement, the answer is:

$$\binom{m-1}{n-1},$$

which in your example is:

$$\binom{3}{1}=3.$$

This is easy to see: set aside an element, say "A". Now pick from the remaining elements. Since order doesn't matter, you get the above binomial coefficient.