Example 4.7 Find the number of non-negative integer solutions to the indeterminate equation $x_{1}+x_{2}+x_{3}=14$ that satisfy the conditions $x_{1} \leqslant 8, x_{2} \leqslant 8$, $x_{3} \leqslant 8$.
Solution: Let the required value be $N$, then $N$ is the coefficient of $\iota^{14}$ in the expansion of
$$
A(t)=\left(1+t+t^{2}+\cdots+t^{8}\right)^{3}
$$

and
$$
\begin{aligned}
A(t) & =\left(\frac{1-t^{9}}{1-t}\right)^{3}=\left(1-t^{9}\right)^{3}(1-t)^{-3} \\
& =\left(1-3 t^{9}+3 t^{18}-t^{27}\right) \sum_{k=0}^{\infty}\binom{k+2}{2} t^{k},
\end{aligned}
$$

Therefore,
$$
\begin{aligned}
N & =\binom{14+2}{2}-3\binom{5+2}{2} \\
& =\binom{16}{2}-3\binom{7}{2}=120-3 \times 21=57
\end{aligned}
$$