When more than two variables are involved, the binomial theorem becomes the multinomial theorem: $$(x_1+x_2+\cdots + x_k)^n=\sum_{p_1+p_2+\cdots+p_k=n}\frac{n!}{p_1!p_2!\cdots p_k!}x_1^{p_1}x_2^{p_2}\cdots x_k^{p_k}$$

Sometimes, the coefficient can be written as $${n\choose p_1, p_2, \cdots, p_k}=\frac{n!}{p_1!p_2!\cdots p_k!}$$

From a counting perspective, the multinomial coeffcient is the number of ways to put $n$ indistinguishable balls to $k$ distinguishable boxes.