Number of Distinct Partial Derivatives of Order n

Theorem
Let $u = \map f {x_1, x_2, \ldots, x_m}$ be a function of the $m$ independent variables $x_1, x_2, \ldots, x_m$.

The number of distinct partial derivatives of $u$ of order $n$ is:
 * $\dfrac {\paren {n + m - 1}!} {n! \paren {m - 1}!}$

That is, the same as the number of terms in a homogeneous polynomial in $m$ variables of degree $n$.