Number of combinations of distributing $n$ balls of $k$ colors to $m$ bins with minimum capacity $c$

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I would like to distribute balls into bins. The balls are indistinguishable in their own color group and the bins are distinguishable and have the same minimum capacity. Here I read that we can take each color as a separate setup and multiply the number of combinations afterwards (please correct me if I'm wrong).

So we should be left with the problem of distributing $n$ indistinguishable balls into $m$ distinguishable bins where each bin has the same minimum capacity of $c$. Another constraint is that every ball has to be distributed to the bins.

I found this question which should be the exact same problem. But when I try out the answer on a simple example I don't get the expected result.

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