find the formula of trinomial expansion

Question

I wonder as if there exist a equivalent forumla to newton binomial

$$(x+y)^n=\sum_{k=0}^{n} {n\choose k} x^{n-k}y^k$$

for three coefficients $(a+b+c)^n$ ?

Answer 1

The expansion is given by $$(a+b+c)^n = \sum_{i,j,k} {n \choose i,j,k}\, a^i \, b^j \, c^k $$ where $n$ is a nonnegative integer and the sum is taken over all combinations of nonnegative indices $i, j$, and $k$ such that $i + j + k = n$. The trinomial coefficients are given by $$ {n \choose i,j,k} = \frac{n!}{i!\,j!\,k!} \,.$$ This formula is a special case of the multinomial formula.