How many nonnegative integer solutions are there to the equation $x_1+x_2+x_3+x_4 = 18$ with restrictions 
How many nonnegative integer solutions are there to the equation 
  $x_1+x_2+x_3+x_4 = 18$ with $$−2\leq x_3 \leq 4\\0 \leq x_2 \leq 5\\3 \leq x_1 \leq8\\ −3 \leq x_4 \leq 13$$

when its in is simple form its easy to solve that but here in this kind of question how i solve it ?  
 A: I never miss a chance to recommend a good book. "How to Count" covers the relevant material and even more and it's a good book indeed. 
Now to the problem. Assuming mistranslating "nonnegative" has a higher likelihood than copy and pasting math expressions, twice (one for "$-2 \leq ...$" and one for "$-3 \leq ...$") I will stick to the following version:

How many integer solutions are there to the equation ...

This class of problems is solved with what is called generating functions. First of all
$$−2\leq x_3 \leq 4 \Leftrightarrow 0 \leq x_3+2 \leq 6$$
$$−3\leq x_4 \leq 13 \Leftrightarrow 0 \leq x_4+3 \leq 16$$
So, with a substitution like $y_1=x_1, y_2=x_2, y_3=x_3+2, y_4=x_4+3$ the problem is equivalent to
$$y_1+y_2+y_3+y_4=23$$
$$3\leq y_1 \leq 8$$
$$0\leq y_2 \leq 5$$
$$0\leq y_3 \leq 6$$
$$0\leq y_4 \leq 16$$
The generating function for this problem is 
$$(y^3+y^4+...+y^8)(y^0+y^1+...+y^5)(y^0+y^1+...+y^6)(y^0+y^1+...+y^{16})$$
the coefficient near the $y^{23}$ term is the answer, which is $232$.
