How to calculate a lot of summation in the same expression

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I'm learning probability and I've just found this on a book:

$$\sum_xP(x)=\sum_{x_1}···\sum_{x_n}P(x_1,...,x_n)=1$$

The bold $x$ is $x=(x_1,x_2,...,x_n)$.

My question is: How can I compute those $\sum_{x_1}...\sum_{x_n}$?

I think the expression means this (multiply each summation):

$$\sum_xP(x)=\sum_{x_1}P(x_1)·\sum_{x_2}P(x_1,x_2)·...·\sum_{x_n}P(x_1,x_2,...,x_n)$$

By the way, I'm trying to calculate the joint probability.