Example 4.13 Find the number of 7-combinations of the multiset $S=\{4 \cdot a, 4 \cdot b, 3 \cdot c, 3 \cdot d\}$.
Solution: Let the required number be $N$, then $N$ is the coefficient of $t^{7}$ in the expansion of
$$
A(t)=\left(1+t+t^{2}+t^{3}+t^{4}\right)^{2}\left(1+t+t^{2}+t^{3}\right)^{2}
$$

and
$$
\begin{aligned}
A(t) & =\left(\frac{1-t^{5}}{1-t}\right)^{2}\left(\frac{1-t^{4}}{1-t}\right)^{2} \\
& =\left(1-t^{5}\right)^{2}\left(1-t^{4}\right)^{2}(1-t)^{-4} \\
& =\left(1-2 t^{5}+t^{10}\right)\left(1-2 t^{4}+t^{8}\right)(1-t)^{-4} \\
& =\left(1-2 t^{4}-2 t^{5}+t^{8}+\cdots\right) \sum_{k=0}^{\infty}\binom{k+3}{3} t^{k},
\end{aligned}
$$

Therefore,
$$
\begin{aligned}
N & =\binom{7+3}{3}-2\binom{3+3}{3}-2\binom{2+3}{3} \\
& =\binom{10}{3}-2 \cdot\binom{6}{3}-2 \cdot\binom{5}{3} \\
& =120-40-20=60 .
\end{aligned}
$$