Suppose a set A = {1,2,3,4,5}.
How would I go about finding the number of sets in the power set of A with a particular cardinality (say 2 for arguments sake)?
I know as A is a particularly small set I could solve this by way of enumerating the power set of A and seeing how many of those sets have a cardinality of 2, but I could I solve this more generally?
The number of subsets of size $k$ of a set of size $n$ is written $\binom{n}{k}$ or ${}^n C_k$, read "$n$ choose $k$", and sometimes called a binomial coefficient. There's a nice formula for it: $$\binom{n}{k} = \frac{n!}{(n-k)!k!}.$$ It can also be read off from Pascal's triangle.
My guess is that you're familiar with some, but not all, of the above material, so I haven't given too many details - they can easily be looked up. And forgive me if I've repeated things you already knew!