(related to Theorem: Multinomial Theorem)
We want to prove that the sum of \(r\) complex or real numbers \(x_1,x_2,\ldots,x_n\) raised to the \(n\)-th power of will expand as follows:
\[(x_1 + x_2 + \ldots + x_r)^n=\sum_{\substack{k_1+\ldots+k_r=n \\ k_1,\ldots,k_r}}\binom{n}{k_1,k_2\ldots,k_r}x_1^{k_1} x_2^{k_2}\ldots x_r^{k_r}.\]
We will prove the theorem by induction on \(r\).
The multinomial theorem takes the form
\[x_1^n=\binom{n}{n}x_1^{n}=x_1^n.\]
Let the multinomial theorem be proved for an \(r\). From the associativity law for complex numbers and by hypothesis, it follows
\[\begin{array}{l}(x_1 + x_2 + \ldots + x_r+ x_{r+1})^n=\\(x_1 + x_2 + \ldots + x_{r-1}+(x_r+ x_{r+1}))^n\\ =\sum_{\substack{k_1+\ldots+k_{r-1}+K=n \\ k_1,\ldots,k_{r-1},K}}\binom{n}{k_1,k_2\ldots,k_{r-1},K}x_1^{k_1} x_2^{k_2}\ldots x_{r-1}^{k_{r-1}}\cdot(x_r+x_{r+1})^K.\end{array}\]
Applying the binomial theorem to the last term in each sum, we get
\[\begin{array}{l}\sum_{\substack{k_1+\ldots+k_{r-1}+K=n \\ k_1,\ldots,k_{r-1},K}}\binom{n}{k_1,k_2\ldots,k_{r-1},K}x_1^{k_1} x_2^{k_2}\ldots x_{r-1}^{k_{r-1}}\cdot(x_r+x_{r+1})^K=\\\sum_{\substack{k_1+\ldots+k_{r-1}+K=n \\ k_1,\ldots,k_{r-1},K}}\binom{n}{k_1,k_2\ldots,k_{r-1},K}x_1^{k_1} x_2^{k_2}\ldots x_{r-1}^{k_{r-1}}\cdot\sum_{k_r}^K\binom K{k_r}x_r^{k_r}x_{r+1}^{K-k_{r}}.\end{array}\]
Note that the following multinomial and the binomial coefficients are equal: \[\binom K{k_r,k_{r+1}}=\binom K{k_r}\quad\quad\text{for }k_r=0,1,\ldots,K\text{ and for }k_{r+1}=K-k_r.\]
Therefore, we can write the last equation also this way:
\[\begin{array}{l}\sum_{\substack{k_1+\ldots+k_{r-1}+K=n \\ k_1,\ldots,k_{r-1},K}}\binom{n}{k_1,k_2\ldots,k_{r-1},K}x_1^{k_1} x_2^{k_2}\ldots x_{r-1}^{k_{r-1}}\cdot(x_r+x_{r+1})^K=\\\sum_{\substack{k_1+\ldots+k_{r-1}+K=n \\ k_1,\ldots,k_{r-1},K}}\binom{n}{k_1,k_2\ldots,k_{r-1},K}x_1^{k_1} x_2^{k_2}\ldots x_{r-1}^{k_{r-1}}\cdot\sum_{k_r+k_{r+1}=K}\binom K{k_r,k_{r+1}}x_r^{k_r}x_{r+1}^{k_{r+1}}.\end{array}\quad\quad ( * )\]
Since, by definition,
\[\binom{n}{k_1,k_2\ldots,k_{r-1},K}\cdot\binom K{k_r,k_{r+1}}=\frac{n ! }{k_1 ! k_2 ! \ldots k_{r-1} ! K ! }\cdot\frac{K ! }{k_r ! k_{r+1} ! }=\frac{n ! }{k_1 ! k_2 ! \ldots k_{r+1} ! }=\binom{n}{k_1,k_2\ldots,k_{r+1}}\]
we get \[( * ) = \sum_{\substack{k_1+\ldots+k_{r+1}=n \\ k_1,\ldots,k_r}}\binom{n}{k_1,k_2\ldots,k_{r+1}}x_1^{k_1} x_2^{k_2}\ldots x_{r+1}^{k_{r+1}}\]
