There is a binomial theorem. How about trinomial theorem?

Question

The binomial theorem is $(a+b)^n = (nC_0)a^n + (nC_1)a^{(n-1)}b^1 + \cdots + (nC_{n-1})a^1b^{(n-1)} + (nC_n)b^n$ Is there such a theorem for $(a+b+c)^n$, or $(a+b+c+d)^n$, or $(a+b+c+d+e)^n$, etc.? If so, what is the proof/reasoning?

Thanks, Selena