Example 9 How many convex $k$-gons are there whose vertices are the vertices of a convex $n$-gon and whose sides are the diagonals of the $n$-gon?
Let the $n$ vertices of a convex $n$-gon be sequentially denoted as $1,2,3, \cdots, n$. The vertices of a convex $k$-gon that meet the conditions are sequentially denoted as $i_{1}, i_{2}, \cdots, i_{k}\left(i_{1}<i_{2}<\cdots<i_{k}\right)$, then there are only the following two possibilities:
(1) $i_{1}=1,3 \leqslant i_{2}<i_{3}<\cdots<i_{k} \leqslant n-1, i_{j+1}-i_{j} \geqslant 2(j=2,3, \cdots, k-1)$.
(2) $2 \leqslant i_{1}<i_{2}<\cdots<i_{k} \leqslant n, i_{j+1}-i_{j} \geqslant 2(j=1,2, \cdots, k-1)$.
In case (1), make the correspondence $\left(i_{2}, \cdots, i_{k}\right) \rightarrow\left(i_{2}-2, i_{3}-3, \cdots, i_{k}-k\right)$, then $1 \leqslant i_{2}-2<i_{3}-$ $3<\cdots \leqslant i_{i-k} \leqslant n-k-1$ are $k-1$ numbers taken from $\{1,2, \cdots, n-k-1\}$, there are $C_{n-k-1}^{k-1}$ ways. Clearly, the above correspondence is a one-to-one correspondence, so in this case, there are $C_{n k-1}^{k-1}$ convex $k$-gons that meet the conditions.

In case (2), make the correspondence $\left(i_{1}, i_{2}, \cdots, i_{k}\right) \rightarrow\left(i_{1}-1, i_{2}-2, \cdots, i_{k}-k\right)$, then $1 \leqslant i_{1}-1<$ $i_{2}-2<\cdots<i_{k}-k \leqslant n-k$ are $k$ numbers taken from $\{1,2,3, \cdots, n-k\}$, there are $C_{n-k}^{k}$ ways. Clearly, this correspondence is also a one-to-one correspondence, so in this case, there are $C_{n k}^{k}$ convex $k$-gons that meet the conditions.

In summary, the number of convex $k$-gons that meet the conditions is $C_{n-k-1}^{k-1}+C_{n-k}^{k}=\frac{n}{n-k} C_{n k-1}^{k-1}$ (when $n<2 k$, $\left.C_{n-k-1}^{k-1}=C_{n-k}^{k}=0\right)$.