The question is how many possible solutions there are for a equation system subject to the condition that $1 \leq x_1 + y_1$, $1 \leq x_2 + y_2$, $1 \leq x_r + y_r$ where $x_i$ and $y_i$ are non-negative integers.
The system is $$x_1 + x_2 + \ldots + x_r = n.$$ $$y_1 + y_2 + \ldots + y_r = n.$$
I want to do it by the inclusion-exclusion principle but i can't find a way to pose a initial equation. After that i would suppose $ 1 - y_1 > x_1 $ and try to introduce it in one of those equations.
Any help is greatly appreciated, thanks.