Generalized permutation and combination: number of solutions How many solutions are there to the equation: $$x_1 + x_2 + x_3 + x_4 + x_5= 21\;,$$
where $x_i$ is a non-negative integer such that 
$$0 \le x_1 \le 3;\; 1 \le x_2 < 4;\text{ and }x_3 \ge 15\;?$$

This should be the coefficient of $a^{21}$ in
$(1+a+a^2+a^3)(a+a^2+a^3)(a^{15}+a^{16}+a^{17}+a^{18}+...)(1+a+a^2+a^3+a^4+...)^2$
where $a$ is some real $0<a<1$
= coefficient of $a^5$ in
$(1+a+a^2+a^3)(1+a+a^2)(1+a+a^2+a^3+a^4+...)^3$
= coefficient of $a^5$ in
$(1-a^4)(1-a^3)(1-a)^{-6}$
= coefficient of $a^5$ in
$(1-a^4 - a^3 + a^7)(1-a)^{-6}$
= coefficient of $a^5$ in
$(1-a^4 - a^3 + a^7)(1+6a+\frac{6.7}{2!}a^2 +\frac{6.7.8}{3!}a^3  +\frac{6.7.8.9}{4!}a^4 + ... )$
$= \frac{6.7.8.9.10}{5!} - 6 - \frac{6.7}{2!}$
