Find the number of integer solutions to the equation $x_1 + x_2 + x_3 = 28$, where $ 3 \leq x_1 \leq 9$, $0 \leq x_2 \leq 8$, and $7 \leq x_3 \leq 17$
I'm having problems with this question.
1) I first tried reducing the range of the variables to $ 0 \leq x_1 \leq 6$,$0 \leq x_2 \leq 8$ and $0 \leq x_3 \leq 10$.
2) That means I have to find the number of integer solutions for $x_1' + x_2' + x_3' = 18$ but I found I cannot reduce the ranges any further.
I have been told to use GPIE (General Principle of Inclusion and Exclusion) in this question but I would like to see other approaches as well.
The answer given is 28.