Example 2-36 Place $n$ identical balls into $m$ labeled boxes, with no empty boxes allowed. How many different schemes are there? Where $n \geqslant m$.


Let the number of ways to place $n$ balls into $m$ boxes be $a_{n}$, and let the generating function of the sequence $\left\{a_{n}\right\}$ be $G(x)$. Since no box can be empty, we have
$$
\begin{aligned}
G(x) & =\underbrace{\left(x+x^{2}+\cdots\right)\left(x+x^{2}+\cdots\right) \cdots\left(x+x^{2}+\cdots\right)}_{m \text { terms }}=\frac{x^{m}}{(1-x)^{m}} \\
(1-x)^{-m} & =1+m x+\frac{m(m+1)}{2!} x^{2}+\cdots+\frac{m(m+1) \cdots(m+k-1)}{k!} x^{k}+\cdots
\end{aligned}
$$

Find the coefficient of $x^{n-m}$, i.e., $k=n-m$.
$$
\begin{aligned}
& \frac{m(m+1) \cdots(m+\overline{n-m}-1)}{(n-m)!}=\frac{m(m+1) \cdots(n-1)}{(n-m)!} \\
= & C(n-1, n-m)=C(n-1, m-1)
\end{aligned}
$$