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In the first part of the question I was asked to find the exponential generating function for $s_{n,r}$, the number of ways to distribute $r$ distinct objects into $n$ (a fixed constant) distinct boxes with no empty box.

I determined the generating function to be $(e^x -1 )^n$. The problem then asks to determine $s_{n,r}$. I'm not sure how this is any different from the first part of the question, but I thought maybe it was asking for a coefficient of $x^r / r!$ How do I do this?

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Employ the following two facts:

$$(e^x-1)^n=\sum_{k=0}^n{n\choose k}e^{kx}(-1)^{n-k},\qquad e^{kx}=\sum_{r\ge0}\frac{k^r}{r!}x^r.$$

The first is from the binomial theorem and second is from the Taylor expansion of $\exp$.

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