Given 's' students in a room and 'd' days in the calendar year, what is the probability 'P' that there will be 'k' "birthday days"?
i.e., 'k = 1' means that everybody's birthday falls on the same day, 'k = 5' means that exactly 5 days of the year coincide with the birthday of at least one student, and 'k = s' means that everyone's birthday is unique.
Assume d ≥ s. 'k' will range from 1 to 's', and ∑P from k = 1 to k = s will be 1 (in other words, P(k) is a probability mass function).
What I Know So Far:
• The total number of ways for the students to have birthdays is d^s.
• P(1) = d/d^(s) (because there are 'd' days on which all students could all share the exact same birthday).
• P(s) = d!/(d^(s)(d-s)!) (because this is compliment of the solution to the standard birthday problem)
Any help/resources that get me closer to a formula for P(k) are greatly appreciated. Thanks!