We all know the birthday problem.
Suppose an inversion* of the birthday problem: given a number of distinct birthdays (c) from a set of people, how can you estimate the number of people you had (n)? (given num birthdays < 365). And for arbitrary bin spaces other than 365 (k)?
E.g. you have c=47 days marked off in your school's calendar as birthdays (k=365), how big is your school most likely (n)?
Birthday collision problems are usually expressed as the odds of getting 1 or more collisions, I think we want the maximally likely quantity of collisions given c and k and add that to c to get a likely n.
(* not sure if this is the right term or name for this question)