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Studying for the math subject GRE, and I come across the titular question. I didn't take any combinatorics or probability courses in college, and I'm realizing I have no intuition for counting. Could someone please explain the solution?

I'd be thankful for any references to resources for a crash course in combinatorics as well, but by no means expected or necessary.

Thanks!

EDIT: And the assumption is the function is $\mathcal{C}^{k}$.

CasaBonita
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  • I suppose they are asking for combinations of partial derivatives with respect to single variables such as $\partial x\partial x\partial y$, that are essentially different, assuming (because of the $\mathcal C^k$ requirement) that they all commute. It is essentially the same as asking for the number of distinct monomials of degree $k$ in $n$ variables. – Marc van Leeuwen Oct 17 '14 at 20:17

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As Marc van Leeuwen said, the partial derivatives are in natural bijection with monomials of degree $k$ in $n$ variables: for example, $$ \frac{\partial^4 f}{\partial x_1 \partial x_3^2\partial x_4} \longleftrightarrow x_1x_3^2x_4 $$

The number of such monomials is $\binom{k+n-1}{n-1}$, as shown by a "stars and bars" argument in this answer.

Community
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