Say I have $n$ balls each of $k$ different colors (i.e. $nk$ balls altogether), and I throw these balls independently into $N$ bins. Is there anything that can be said (in expectation, limits with respect to any of the variables, or otherwise) about the minimum number of bins that collectively contain at least one ball of each color? I can't seem to find this problem in the literature, although it seems natural to me. How about the case where $N = nk$?

To clarify, I am interested in the question "how many bins do I have to look in to see one ball of every color", pursuant to Ben Barber's comment.