Combinatorics problem with negative possibilities I know how to solve the basic number of solutions equations, such as "find the number of positive integer solutions to $x_1 + x_2 + x_3$ = 12, with ". But I have no clue how to do this problem:
Find the number of solutions of $x_1+x_2-x_3-x_4 = 0$ in integers between -4 and 4, inclusive.
If I try and solve it like the basic equations, I get $C(n+r-1,r)$$ = C(0+9-1,9)$$ = C(8,9)$, which is obviously improper. Can someone point me in the right direction on how to solve this type of problem?

One may also solve this using generating functions.  
Following the others' suggestion to consider the equation $y_1 + y_2 + y_3 + y_4 = 16$, you would want to find the coefficient of $x^{16}$ in the product $(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8)^4$, which a computer algebra system can easily compute.
The general principle of using generating functions for combinatorial problems is simple.  Suppose you want to find the number of solutions to
$$
\sum_{i=1}^n x_i = A,
$$
where $a_i \leq x_i \leq b_i$ for $i = 1, \dots n$.  Then, the number of solutions to the equation is the same as the coefficient of $x^A$ in the product
$$
(x^{a_1} + \dots + x^{b_1})(x^{a_2} + \dots + x^{b_2}) \dots (x^{a_n} + \dots + x^{b_n}).
$$
The reason this works is that finding a term in the product above which multiplies to $x^A$ amounts to finding $x^{c_1}, x^{c_2}, \dots, x^{c_n}$ where $a_i \leq c_i \leq b_i$ and $x^{c_1 + c_2 \dots + c_n} = x^A$.  The advantage of the generating function approach is that computer algebra systems (like mathematica or sage) can easily handle polynomial multiplication.
