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$\dbinom{n+r−1}{r−1}$ is the answer for the distribution of $n$ identical objects among $r$ persons. Not for the groups, because groups are considered as identical it do not have name. Ex. two identical balls can to be distributed among two persons $[(2,0)(0,2)(1,1)]$ in three ways. But when we go for groups $(2,0)$ and $(0,2)$ considered as the same, so now the answer will be only two.

What will be the formula if we want to remove the same groups ex. in the above example if we want $(2,0)$ and $(0,2)$ to be considered same and want the answer $2$ so what will be the formula in that case?

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1 Answer

There are the partitions of $n$ into $r$ or fewer parts.

There is no pleasant "closed form" formula.

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