sigma notation confusion

Question

Can someone please explain to me what this sigma notation means in this formula. Im confused because there is no top number on the sigma and what does this sum mean on the bottom? and also this is the formula for the multinomial theorem, what are the $k_m$ values supposed to represent? $$(x_1+x_2+...+x_m)^n=\sum_{k_1+k_2+...+k_m=n}{n \choose k_1,k_2,...,k_m}\prod_{t=1}^mx_t^{k_t}$$

Answer 1

$\sum_{k_1+k_2+...+k_m=n}$ means you're considering the sum over all elements of the following set $$\bigg\{(k_1,k_2,\ldots,k_m):\sum_{i=1}^mk_i=n,0\leq k_i\leq n\bigg\}$$

Answer 2

The $k_i$ are nonnegative integers whose sum is $n$. $k_i$ is the exponent of $x_i$ in a typical term in the expansion of $(x_1+\dots+x_m)^n$ The sum is taken over all $m-$tuples $(k_1,\dots,k_m)$ such that $k_1+\cdots+k_m=n.$

Answer 3

Here is an application: $$(x_1+x_2+x_3)^4=\sum_{k_1+k_2+k_3=4}{4 \choose k_1,k_2,k_3}\prod_{t=1}^3x_t^{k_t}=\\ {4\choose 4,0,0}x_1^4x_2^0x_3^0+{4\choose 3,1,0}x_1^3x_2^1x_3^0+{4\choose 3,0,1}x_1^3x_2^0x_3^1+{4\choose 2,2,0}x_1^2x_2^2x_3^0+\\ {4\choose 2,1,1}x_1^2x_2^1x_3^1+{4\choose 2,0,2}x_1^2x_2^0x_3^2+{4\choose 1,3,0}x_1^1x_2^3x_3^0+{4\choose 1,2,1}x_1^1x_2^2x_3^1+\\ {4\choose 1,1,2}x_1^1x_2^1x_3^2+{4\choose 1,0,3}x_1^1x_2^0x_3^3+{4\choose 0,4,0}x_1^0x_2^4x_3^0+{4\choose 0,3,1}x_1^0x_2^3x_3^1+\\ {4\choose 0,2,2}x_1^0x_2^2x_3^2+{4\choose 0,1,3}x_1^0x_2^1x_3^3+{4\choose 0,0,4}x_1^0x_2^0x_3^4=\\ x_1^4 + 4 x_2 x_1^3 + 4 x_3 x_1^3 + 6 x_2^2 x_1^2 + \\ 6 x_3^2 x_1^2 + 12 x_2 x_3 x_1^2 + 4 x_2^3 x_1 + 4 x_3^3 x_1 + \\ 12 x_2 x_3^2 x_1 + 12 x_2^2 x_3 x_1 + x_2^4 + x_3^4 + \\ 4 x_2 x_3^3 + 6 x_2^2 x_3^2 + 4 x_2^3 x_3,$$ Wikipedia answer.

Note: The order of terms differs. Also: ${4\choose 3,1,0}=\frac{4!}{3!1!0!}=4$.