Combinatorics: Number of integer solutions with lower bounds

Question

How many integer solutions does the equation $x_1 + x_2 + x_3 + x_4 = 15$ have, if we require that $x_{1}\ge 2, x_{2}\ge 3, x_{3}\ge 10, x_{4}\ge −3?$

I need to understand in general what to do. I don't know the answer so can't check myself.

Answer 1

The problem is the same than to find the number of solution of $$x_1+x_2+x_3+x_4=3,$$ where $x_1,...,x_4\geq 0$. And this is a well know problem which it refer to the stars and bars problem. So, there are $\frac{6!}{3!3!}$ solutions.