How many total colors in the urn, given that I've seen C colors in M draws?

Question

In this problem I want to estimate the unknown total number of colors of balls in the urn. Say that there are N total balls in the urn, and after m draws (with replacement, if that makes it easier), I've observed k colors. We can assume that all colors are equally represented. N can be >> m, N >> k, which may also simplify. Extreme cases: If I've drawn 5 times and see blue,blue,blue,blue,blue, then it's likely that all the balls are blue and there are no other colors. But if I draw 5 times and see red,orange,yellow,green,blue, then I have no idea how many colors there could be -- up to N. My conjecture is that the answer is mk/(m - k) for N >> m but I'd appreciate a proper derivation.