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Is there any derivation for the problems:

$1.$Dividing $n$ identical things into $r$ identical groups where some groups may be empty or all $n$ things can go to any one group.

$2.$Dividing $n$ identical items into $r$ identical groups where there is at-least one item in every group.

I am not sure if this is modeled as some other known problem,what I am looking for, is a nice approach/derivation for solving these kind of models.

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I think you are refering to partitions and compositions.

Edited: The restriction of "r groups" introduce a restriction. The second problem would correspond to counting $p(n,r)$, number of partitions of a number $n$ in $r$ parts. This is not trivial (see here or here, our counting corresponds to $p(n,k)$ there). The first problem would correspond to the number of partions of size up to $r$, hence it can be expressed as a sum $\sum_{k=1}^r p(n,k)$

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    I agree that this is not trivial,thanks for the infos. :-)2011-09-03