Given the following problem:
A national singing contest has five distinct entrants from each state. Use a generating function for modeling the number of ways to pick $20$ semifinalists if there are at most three people from each state.
Up until this question, I've been dealing with only identical objects. I'd like to say that the function will be
but the problem with this is that I'm assuming that all of the contestants are identical. How can I take account for the fact that each contestant is distinct?