Example 4 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 convex $n$-gon be $A_{1} A_{2} \cdots A_{n}$, and a convex $k$-gon that meets the conditions be $A_{i_{1}} A_{i_{2}} \cdots A_{i_{k}}\left(1 \leqslant i_{1}<\right.$ $\left.i_{2}<\cdots<i_{k} \leqslant n\right)$. 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), we establish the correspondence $\left(i_{2}, i_{3}, \cdots, i_{k}\right) \rightarrow\left(i_{2}-2, i_{3}-3, \cdots, i_{k}-k\right)$, then $1 \leqslant i_{2}-2<i_{3}-$ $3<\cdots<i_{k}-k \leqslant n-k-1$ are $k-1$ numbers chosen from $\{1,2, \cdots, n-k-1\}$, with $C_{n-k-1}^{k-1}$ different ways. Clearly, this correspondence is one-to-one. Therefore, in this case, there are $C_{n-k-1}^{k-1}$ convex $k$-gons that meet the conditions.

In case (2), we establish 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 chosen from $\{1,2, \cdots, n-k\}$, with $C_{n-k}^{k}$ different ways. This correspondence is also one-to-one, 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 $C_{n-k-1}^{k-1}+C_{n-k}^{k}=\frac{n}{k} C_{n-k-1}^{k-1}$ (when $n<2 k$, $C_{n-k-1}^{k-1}=C_{n-k}^{k}=0$).