Here is an approach that may work. Maybe you can answer these questions:
1. There are three types of months: those with no birthdays (4 of them), those with 2 birthdays (4 of them), and those with 3 birthdays (4 of them). How many ways can you assign months to those three classes?
2. Given an assignment of months to those classes, in how many ways can 20 people be assigned to the months?