Find the number of solutions to the equation $x_1+x_2+x_3+x_4=19$ with $0\leq x_i\leq 8$.
I know that I should use inclusion-exclusion, but I don't quite see why.
If I had this problem:
Find the number of solutions to the equation $x_1+x_2+...+x_5=10$ with no restrictions to $x_i$:
The solution to this would be $14 \choose 10$ (like a stars-bars problem).
Back to the first problem, I see why can't use that... Let's say I want to solve something equivalent such as:
$(x_1+8)+(x_2+8)+(x_3+8)+(x_4+8)=19$ with no restrictions to $x_i$.
That would be $x_1+x_2+x_3+x_4=-13$ which doesn't make sense as I'm working with natural numbers.
Can someone explain me why inclusion-exclusion applies to this? I understand the theorem but I don't get why I should use it on this.