【Example 2】Find the coefficient of the term $a_{1}^{k_{1}} a_{2}^{k_{2}} \cdots a_{m}^{k_{m}}$ in the expansion of $\left(a_{1}+a_{2}+\cdots+a_{m}\right)^{n}$, where $k_{1}+k_{2}+\cdots+k_{m}=n$.
When $m=2$, by the binomial theorem, the coefficient of $a_{1}^{k_{1}} a_{2}^{n-k_{1}}$ is $C_{n}^{k_{1}}$. From the perspective of permutations of non-identical elements, since
$$
\left(a_{1}+a_{2}\right)^{n}=\underbrace{\left(a_{1}+a_{2}\right)\left(a_{1}+a_{2}\right) \cdots\left(a_{1}+a_{2}\right)}_{n \uparrow}
$$

the expansion of each term is the product of several $a_{1}$ and several $a_{2}$, and the number of terms with the value $a_{1}^{k_{1}} a_{2}^{n-k_{1}}$ is exactly the number of all possible permutations of $k_{1}$ $a_{1}$ and $n-k_{1}$ $a_{2}$, so the coefficient of this term should be

According to this perspective, when $m=3$, the coefficient of the term $a_{1}^{k_{1}} a_{2}^{k_{2}} a_{3}^{k_{3}}=a_{1}^{k_{1}} a_{2}^{k_{2}} a_{3}^{n-k_{1}-k_{2}}$ in the expansion of $\left(a_{1}+a_{2}+a_{3}\right)^{n}$ should be

In general, the coefficient of the term $a_{1}^{k_{1}} a_{2}^{k_{2}} \cdots a_{m}^{k_{m}}=$ $a_{1}^{k_{1}} a_{2}^{k_{2}} \cdots a_{m}^{n-k_{1}-k_{2}-\cdots-k_{m-1}}$ in the expansion of $\left(a_{1}+a_{2}+\cdots+a_{m}\right)^{n}$ should be
$$
\begin{aligned}
\left(\begin{array}{cccc}
n & & \\
k_{1} & k_{2} & \cdots & k_{m}
\end{array}\right) & =\frac{n!}{k_{1}!k_{2}!\cdots k_{m}!} \\
& =\frac{n!}{k_{1}!k_{2}!\cdots\left(n-k_{1}-k_{2}-\cdots-k_{m-1}\right)!}
\end{aligned}
$$