CDF and density of the kth smallest of n draws

Let $X_1,\dots,X_n$ be i.i.d. with CDF $F$ and density $f$, and let $X_{(k)}$ be the $k$-th smallest of them. Find the CDF and the density of $X_{(k)}$.

$\mathbb{P}(X_{(k)}\le x)=\sum_{j=k}^{n}\binom nj F(x)^j(1-F(x))^{n-j}$, with density $\dfrac{n!}{(k-1)!\,(n-k)!}F(x)^{k-1}(1-F(x))^{n-k}f(x)$. The CDF counts the ways at least $k$ draws fall below $x$. For the density, one draw sits at $x$, $k-1$ fall below and $n-k$ above.