Extending the Binomial Expansion to More Variables

We can general the binomial expansion, in which the coefficient of  
\[a^xb^{n-x}\]
  in the expansion of  
\[(a+b)^n\]
  is  
\[{}^{n}C_x= \frac{n!}{x!(n-x)!}\]
  to any number of variables. Suppose we call the variables  
\[\{ x_1,x_2,...,x_k \}\]
. For  
\[(x_1+x_2+...+x_k)^n\]
  the coefficient of  
\[x_1^{m_1}x_2^{m_2}...x_k^{m_k}\]
  is  
\[\frac{n!}{(m_1)!(m_2)!...(n_k)!}\]
.
Example:
\[(p+q+r)^4\]

\[\frac{4!}{4!0!0!}=1, \; \frac{4!}{3!1!0!}=4{}, \; \frac{4!}{2!2!0!}=1^6, \; \frac{4!}{2!1!1!}=12\]
.
All the coefficients are symmetric with respect in interchange of factors in the denominator. The expansion is
\[\begin{equation} \begin{aligned} (p+q+r)^4 & =p^4+q^4+r^4 \\ &+ 4p^3q+4p^3r+4pq^3+4q^3r+4pr^3+4qr^3 \\ &+ 6p^2q^2+6p^2r^2+6q^2r^2 \\ &+12p^2qr+12pq^2r+12pqr^2 \end{aligned} \end{equation}\]

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