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I have a question that I have not been able to find the answer to. Suppose I have balls of $k$ different colors (the balls of each color indistinguishable), $a(j)$ of each color for $j \le k$ . How many ways can I distribute those balls (i.e., the sum of the $a(j)$ ) into $n$ distinguishable bins? Most balls to bins problems assume that there is only one color, sometimes limiting the maximum number of balls allocated to a given box.

I have looked at binomial type generating functions to no avail (for the general case), and the "twelvefold way" stars and bars approach, but no luck. I understand the Polya/Burnside approach might work, but haven't been able to formulate the problem precisely. Can anyone provide solution or point me in the right direction?

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    Are you requiring that each bin receive at least one color, or is there some other constraint?2012-04-11

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