10. Let $A=\{1,2, \cdots, n\}(n>1, n \in \mathbf{N})$, mapping $f: A \rightarrow A$. Then the number of $f$ that satisfy $f(1) \leqslant f(2) \leqslant \cdots \leqslant f(n)$, and the image takes exactly $k(1<k \leqslant n)$ different values is $\qquad$
10. $\mathrm{C}_{n}^{k} \mathrm{C}_{n-1}^{k-1}$.

First, select $k$ images from $1 \sim n$, there are $\mathrm{C}_{n}^{k}$ ways to do this.
Then divide $1 \sim n$ into $k$ segments, with at least one number in each segment, i.e., sequentially select $k-1$ numbers $a_{1}, a_{2}, \cdots, a_{k-1}$ from the first $n-1$ numbers, then $a_{i-1}+1$ to $a_{i}$ is the $i$-th segment $(i=1,2, \cdots, k), a_{0}=0, a_{k}=n$. There are $\mathrm{C}_{n-1}^{k-1}$ ways to do this.
Therefore, there are $\mathrm{C}_{n}^{k} \mathrm{C}_{n-1}^{k-1}$ functions $f$ that satisfy the condition.