# How do I find a closed form expression for a sum

If $$k$$ and $$n$$ are positive integers, how do I give a simple closed form expression for the sum $$\sum_{a_1+···+a_k=n} {n \choose a_1,...,a_k}$$

I'm not sure the process of finding a closed form expression and I think understanding this general case will help for explicit questions like $${9 \choose a, b, c}$$

By the multinomial theorem, the sum in question evaluates to $$(1 + 1 + \cdots + 1)^n = k^n$$. For $$k = 2$$ this reduces to the famous binomial identity $$\sum_a \binom n a = 2^n$$.
More explicitly: For $$k = 2$$ we get $$\sum_a \binom n a = \sum_a \binom n a 1^a 1^{n-a} = (1+1)^n = 2^n$$. And for $$k = 3$$ we get $$\sum_{a_1+a_2+a_3 = n} \binom {n} {a_1,a_2,a_3}1^{a_1} 1^{a_2} 1^{a_3} = (1+1+1)^n = 3^n$$.
• so does that mean the closed form of i.e. $\sum_{a+b+c=7} {7 \choose a, b, c}$ is $(1+...+1)^7$ ? – user3427042 Apr 26 at 6:30
• Yes, with $3$ $1$'s. So $3^7$. – Yakov Shklarov Apr 26 at 6:31