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The number of ways in which $n$ different things can be distributed into $r$ different groups is $$r^n - \binom{r}{1} (r - 1)^n + \binom{r}{2} (r - 2)^n + \ldots + (- 1)^{r - 1} \binom{r}{r - 1}$$.

This is what my book wrote. Now, how to prove this? Also, as the title goes, what is the difference between distribution & arrangement?

  • Are you considering both the "things" and the "groups" to be distinguishable? ${}\qquad{}$ – Michael Hardy Apr 26 '15 at 16:48
  • There is no $x$ in your formula – Marc Apr 26 '15 at 16:48
  • It is an InclusionExclusion argument. we are counting the functions from an $n$-element set onto an $r$-element set. Please see the Wikipedia article on Stirling numbers of the second kind. There have also been a number of relevant solutions on MSE. – André Nicolas Apr 26 '15 at 16:52

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From the formula, it seems that by a distribution $d(n,r)$ it means partitioning an n-element set into r-sets such that none of the r-sets are empty.

Also, in that case $d(n,r)$ is the number of onto functions from a n-element set to a r-element set and you can find the derivation here.

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