Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 1 down vote accepted

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 $\sum_{k=1}^r p(n,k)$

share|cite|improve this answer
May be related but I think not exactly as there is a restriction of $r$ which needs to be considered explicitly?! – Quixotic Aug 21 '11 at 17:46
YOu are right, added info. – leonbloy Aug 21 '11 at 20:22
I agree that this is not trivial,thanks for the infos. :-) – Quixotic Sep 3 '11 at 15:44

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.