Number theory problem

noospace

Hi all,

Consider the the number of distinct permutations of a collection of [latex]N[/latex] objects having multiplicities [latex]n_1,\ldots,n_k[/latex]. Call this F.

Now arrange the same collection of objects into [latex]k[/latex] bins, sorted by type. Consider the set of permutations such that the contents of any one bin after permutation are the same.

Can anyone help to convince me that the number of permutations which achieve this is also F? I believe that this is probably true but I'm unable to show it.

I've read elsewhere that [latex]F = N!/(n_1!\cdots n_k!)[/latex] which provides a starting point, but I'm not sure where to go from here.