This problem seems to be the following variation of the birthday problem: what is the probability that in a group of $p$ people nobody has a birthday that $<k$ people in the group have.
I'm also interested in the opposite problem, where nobody has a birthday that $\geq k$ people in the group have.
I think the exact probability can be calculated using the inclusion-exclusion principle, but does a simpler approach (I mean an expression that is easier to compute) or decent approximation exist here?