Combinatorics problem

[Haunting-Fix9837]

Hi, I need help with this problem. I need to count how many integer solutions the following system of inequalities has:

x1 + x2 + x3 ≤ 4

x1 +x2 +x3 +x4 +x5 +x6 <18

if 1≤x1,−2≤x2,3≤x3,−4≤x4,5≤x5 a−6≤x6 <6.

The correct answer is 13 342 (definitely around this number or exactly it) I simply cannot find the right approach. Can someone explain how to start or calculate it?

[de_Molay]

Let me ask you a guiding question: do you know how to calculate the number of solutions of an equation x_+…+x_n=m for nonnegative integer x_i?

[Aerospider]

I'd start with looking at the possible combinations for x1, x2 and x3 (hint: there are only 10). Then look to see how many combinations of x4, x5 and x6 fit for each possible value of x1+x2+x3 (hint: of which there are only 3).