How many ways can $N$ labelled balls be placed in $M$ unlabelled boxes, provided each box must have at least $P$ balls inside?
Naturally $N > M \times P$.
Any closed form solutions would be great!
The motivation behind this question is that I want to brute force sample each of the possible configurations for particular values of $N$, $M$ and $P$. However, given that the quantity may be fairly large I want to check how many there are before trying!