From a discussion with some friends, this *apparently easy* problem has come out; I decided to post it here, because I believe that the answer is non-trivial and the maths beneath interesting. Partial solutions, ideas or possible approaches are welcome too!

Suppose that there are $P$ boxes (of infinite capacity) and that every second, I choose a box uniformly at random and I put a bean in it.

1) How many seconds $T$ are at least necessary to have a probability greater than $q \in [0,1]$ of having at least $N$ beans in each box?

2) How many seconds $T$ are at least necessary to have a probability greater than $q \in [0,1]$ of having at least $N$ beans in a fraction at least $f \in [0,1]$ of the total number $P$ of boxes?

I apologize for the "imaginative" formulation of my question, but I hope this choice makes the problem clearer.