The problem is:
Find the number of integer solutions to the equation $$ x_1 + x_2 + x_3 + x_4 = 15 $$ satisfying $$ \begin{align} 2 \leq &x_1 \leq 4, \\ -2 \leq &x_2 \leq 1, \\ 0 \leq &x_3 \leq 6, \text{ and,} \\ 3 \leq &x_4 \leq 8 \>. \end{align} $$
I have read some papers on this question, but none of them explain clearly enough. I am especially confused when you must decrease the total amount of solutions to the equation—with no regard to the restrictions—from the solutions that we don't want. How do we find the intersection of the sets that we don't want? Either way, in helping me with this, please explain this step.