I am currently facing the following problem:
There is a set of integers from 1 to Y, and N people choosing a random number of this set. What is then the probability that there is at least one number which is chosen at least X times?
For X = 2 (so in other words, there is no double occupancy at all), I know this problem can be solved using
$P = 1 - \frac{k!}{k^n (k-n)!}$,
as explained in the answer to this question: Probability of choosing the same number
However, I am unable to come up with a solution to the more general form as in my question. Can anyone think of a general method of determining this probability?
