generating functions / combinatorics Calculate number of solutions of the following equations:
$$ x_1 + x_2 + x_3 + x_4 = 15 $$
where $ 0 \le x_i < i + 4 $ 
I try to solve it using generating functions/enumerators :
$$ (1+x+x^2+x^3+x^4)(1+x+x^2+x^3+x^4+x^5)(1+x+x^2+x^3+x^4+x^5+x^6)(1+x+x^2+x^3+x^4+x^5+x^6+x^7)$$
and take coefficient near $15$. But I do not know how to quickly calculate it. Maybe there exists any faster way?

To take a slightly different route from i707107's solution, once you obtain the product of rational functions, you can use the identity:
and 
$$
\frac{1}{(1-x)^n} = 1  + \binom{1 + n -1}{1}x  + \binom{2 + n -1}{2}x^2 + \dots +  \binom{r + n -1}{r}x^r + \dots.
$$
to expand the last term of the simplified product (2nd line) that i707107 has written. Then you would simply take product of the first $4$ polynomials, which isn't that bad since they're just $2$ terms each. Then you'd find the coefficient of the products that give $x^{15}$.
