Number of Integer Solutions for $x_1 + x_2 + x_3 + x_4 = 15$ where $-5 \le x_i \le 10$

Question

I am trying to find the number of Integer Solutions for $x_1 + x_2 + x_3 + x_4 = 15$ where $-5 \le x_{i_{\in [4]}} \le 10$

I know if $x_i$s are all non-negative integers, it is a number partition of 15 however, this case a bit tricky with the possible negative integers.

Any hint to pinch on this problem?

Answer 1

Let $y_i = x_i + 5$ for each $i$. Then you're trying to find the number of integer solutions to $y_1 + y_2 + y_3 + y_4 = 35$ with $0 \le y_i \le 15$.