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Find an exponential generating function for the number of distributions of $r$ distinct objects into $n$ different boxes w/ exactly $m$ nonempty boxes

I'm not sure about the solution, but this is what I obtained after using some identities: $$ (e^x - 1)^m (1)^{n-m}. $$ Is this correct?

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The question is not clear: you have $r$, $n$ and $m$ in the question; do you want the generating funcion in three variables, or do you want to keep (say) $n$ and $m$ fixed and consider the generating function's exponents to mark $r$? –  ShreevatsaR Feb 6 at 9:12

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