In how many ways can 4 unlabeled blue balls and 2 unlabeled red balls be packed in 5 different boxes?

combinatorics

I have 4 unlabeled (identical) blue balls and 2 identical redz balls and 5 different (distinct) boxes. In how many ways all balls can be placed in these 5 boxes?


I will assume that you know how to count the number of ways to distribute $n$ identical candies among $r$ children. Compute this number for the blue balls, call it $B$. Compute this number for the red balls. Call it $R$. For every way of distributing the blue balls, there are $R$ ways to distribute the red balls, so our total is $BR$.

If you do not know how to solve the general $n$ candy $r$ child problem, I suggest you look at the Wikipedia article on Stars and Bars. The answer turns out to be $\binom{n+r-1}{r-1}$. The reasoning is well-described in the Wikipedia article, and in several MSE answers.

For our particular very small numbers, $B$ and $R$ can also be computed by making careful lists.

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