Example 2-25 $n$ indistinguishable balls are placed into $m$ indistinguishable boxes, with no box left empty. How many ways are there to do this?
where $n \geqslant m$.
The generating function for the number of solutions $a_{n}$ is
$$
G(x)=\underbrace{\left(x+x^{2}+\cdots\right)\left(x+x^{2}+\cdots\right) \cdots\left(x+x^{2}+\cdots\right)}_{m \text { terms }}=\frac{x^{m}}{(1-x)^{m}}
$$

Since $(1-x)^{m}=1+m x+\frac{m(m+1)}{2!} x^{2}+\frac{m(m+1)(m+2)}{3!} x^{3}+\cdots$
the coefficient of the $x^{\prime \prime}$ term is
$$
\begin{aligned}
\frac{m(m+1)(m+2) \cdots(m+n-m-1)}{(m-1)!(n-m)!} & =\frac{m(m+1) \cdots(n-1)}{(n-m)!} \\
& =\frac{(n-1)!}{(m-1)!(n-m)!}=C(n-1, m-1)
\end{aligned}
$$