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I have a problem that hopefully belongs in this forum.

Suppose that I have a bag full of M marbles, each marble being one of N colors, N =< M. The number of marbles of color = N is p_n, so the sum over n of p_n equals M. Next I want to arrange all of my marbles in order from 1 to M. The question: how many unique ways are there to do this, as a function of the variables p_n?

I know for example that if each marble is its own separate color, ie M = N, then there are exactly M! different ways to arrange them. Alternatively if there is only one color, ie N = 1, then there is only one way to arrange them. It is the more general case that I have not yet solved.

David