多项式定理
\[\because (a_1+a_2+...+a_n)^m=\sum_{r_1+r_2+...+r_n=m} C_m^{r_1}C_{m-r_1}^{r_2}...C_{r_n}^{r_n} a_1^{r_1}a_2^{r_2}...a_n^{r_n} \quad (根据每个字母项对应的次数及系数的分配可得)
\]
\[\because C_m^n=\frac{m!}{n!(m-n)!}
\]
\[\therefore C_m^{r_1}C_{m-r_1}^{r_2}C_{m-r_1-r_2}^{r_3}...C_{r_n}^{r_n}=\frac{m!}{{r_1}!(m-r_1)!}\frac{(m-r_1)!}{{r_2}!(m-r_1-r_2)!}\frac{(m-r_1-r_2)!}{{r_3}!(m-r_1-r_2-r_3)!}...\frac{(m-r_1-r_2-...-r_{n-1})!}{r_n!(m-r_1-r_2-...-r_n)}
\]
\[\because r_1+r_2+...+r_n=m
\]
\[\therefore C_m^{r_1}C_{m-r_1}^{r_2}C_{m-r_1-r_2}^{r_3}...C_{r_n}^{r_n}=\frac{m!}{r_1!r_2!...r_n!}
\]
\[\therefore (a_1+a_2+...+a_n)^m=\sum_{r_1+r_2+...+r_n=m} \frac{m!}{r_1!r_2!...r_n!}a_1^{r_1}a_2^{r_2}...a_n^{r_n}
\]