# Case Control Study

Suppose we have identified $20$ cases from a population of $1000$ people. $15$ of these cases are male. We want to use one-to-one matching to choose the control group. We would match on gender. So would we just randomly choose $15$ males from the population that are not cases and then arbitrarily match them to the $15$ cases? There are $\binom{15}{2}$ matchings possible.

That's right, although of course there are plenty of other things you could match on as well as gender. And I didn't get your calculation of the number of matchings possible: there are $\frac{n!}{(n-15)!}$ ways to match the male cases, where $n$ is the number of males in the population that are not cases.