($x_1$ + $x_2$ + ... + $x_k$)^n = sum over all ($r_1$,...,$r_k$) with sum $r_i$ = n of [n!/($r_1$!$r_2$!...*$r_k$!)] * $x_1^{r_1}$ * $x_2^{r_2}$ * ... * $x_k^{r_k}$.

Number of terms: C(n+k-1, k-1). For trinomial (k=3): C(n+2, 2) terms.

The coefficient n!/($r_1$!...$r_k$!) is the multinomial coefficient. It counts the number of ways to arrange n objects where $r_i$ are of type i.

Application: Finding coefficient of $x^a$$y^b$$z^c$ in (x+y+z)^n = n!/(a!*b!*c!) if a+b+c = n.