Example 4.11 Select $k$ elements from an $n$-element permutation to form a combination, such that any two elements $a$ and $b$ in the combination satisfy the condition: in the original permutation, there are at least $r$ elements between $a$ and $b$ (with $n+r \geqslant k r+k$). Let $f_{r}(n, k)$ denote the number of different combinations that can be formed. Find $f_{r}(n, k)$.
Solution: Let's assume the given $n$-element permutation is $\pi=123 \cdots n$. From $\pi$, we select $k$ elements that satisfy the problem's conditions as $a_{1}, a_{2}, \cdots, a_{k}\left(a_{1}<a_{2}<\cdots<a_{k}\right)$. Suppose in $\pi$, there are $x_{1}$ elements before $a_{1}$, $x_{k+1}$ elements after $a_{k}$, and $x_{i+1}$ elements between $a_{i}$ and $a_{i+1}(i=1,2, \cdots, k-1)$. Then $x_{1} \geqslant 0, x_{i+1} \geqslant r(i=1,2, \cdots, k-1), x_{k+1} \geqslant 0$ and $x_{1}+x_{2}+\cdots+x_{k+1}=n-k$. Therefore, $f_{r}(n, k)$ is the number of non-negative integer solutions to the equation $x_{1}+x_{2}+\cdots+x_{k+1}=n-k$ that satisfy the condition $x_{i+1} \geqslant r(i=1,2, \cdots, k-1)$, thus $f_{r}(n, k)$ is the coefficient of $t^{n-k}$ in the expansion of
$$
A(t)=\left(1+t+t^{2}+\cdots\right)^{2}\left(t^{r}+t^{t+1}+t^{r+2}+\cdots\right)^{k-1}
$$

and
$$
\begin{aligned}
A(t) & =(1-t)^{-2} t^{(k-1) r}(1-t)^{-(k-1)} \\
& =t^{k r-r}(1-t)^{-(k+1)} \\
& =t^{k r-r} \sum_{j=0}^{\infty}\binom{k+j}{k} t^{j},
\end{aligned}
$$

Let $k r-r+j=n-k$, then $k+j=n-k r+r$, so
$$
f_{r}(n, k)=\binom{n-k r+r}{k} .
$$