Number of particular terms in a product I hope this question is not too stupid, but I know very little about combinatorics and i can't find an answer..

So fix $r\in \mathbb{N}$. Let $x_1,\ldots, x_r$ be variables, let the product between them be commutative and consider the quantity $(x_1+\ldots+x_r)^{2r-2}$. Of this i want to consider only the terms of the form $x_1^2x_2^2\ldots x_{i-1}^2 x_{i+1}^2\ldots x_r^2$, i.e. the terms in which the variable $x_i$ doesn't appear and all the other variables $x_j$ appear with power exactly 2. Of course, because the product is commutative, consider all the permutations.

Fixed $i$ between $1$ and $r$, how many terms of this said form $x_1^2x_2^2\ldots x_{i-1}^2 x_{i+1}^2\ldots x_r^2$ are there? the answer should be $\frac{(2r-2)!}{2^{r-1}}$, but i can't figure out why, would you enlighten me?
Here's a very useful and general fact. Consider a product:

$$(a_1+a_2+...)(b_1+b_2+...)...$$

There are a finite number of factors, and a finite number of terms within each factor, but I'll write it as above to keep the notation trimmer. Let's say there are $n$ factors. I'll use the term $n$-product to refer to a product of $n$ arbitrary numbers. Then the above product is equal to the sum over all possible $n$-products that can be formed by taking one term from inside each set of parentheses. For example:

$$(a+b)(c+d+e)=ac+ad+ae\quad\leftarrow\text{terms where we chose $a$ from the first factor}\\
+bc+bd+be\quad\leftarrow\text{terms where we chose $b$}$$
Here's a very useful and general fact. Consider a product:

$$(a_1+a_2+...)(b_1+b_2+...)...$$

There are a finite number of factors, and a finite number of terms within each factor, but I'll write it as above to keep the notation trimmer. Let's say there are $n$ factors. I'll use the term $n$-product to refer to a product of $n$ arbitrary numbers. Then the above product is equal to the sum over all possible $n$-products that can be formed by taking one term from inside each set of parentheses. For example:

$$(a+b)(c+d+e)=ac+ad+ae\quad\leftarrow\text{terms where we chose $a$ from the first factor}\\
+bc+bd+be\quad\leftarrow\text{terms where we chose $b$}$$