Counting fifth-order partials in ten variables

Let $f(x_1,\dots,x_{10})$ be a real-valued function on $\mathbb{R}^{10}$ with continuous partial derivatives of every order, so mixed partials do not depend on the order of differentiation. How many distinct partial derivatives of order $5$ does $f$ have? (For instance, $\partial^5 f/\partial x_3^5$ and $\partial^5 f/\partial x_1\partial x_4^2\partial x_9^2$ each count as one.)

$2002$. A fifth-order partial is determined by how many times each variable is differentiated, i.e. by a non-negative solution of $n_1+\dots+n_{10}=5$. Stars and bars gives $\binom{14}{5}=2002$.