4. (10 points) [O3] How many solutions are there to the equation
$x_1 + x_2 + x_3 + x_4 = 20$
where $x_i$, $i = 1, 2, 3, 4$ are non-negative integers such that
(a) $x_i \ge 2$ for all $i$?
(b) $x_1 \ge 2$, $x_2 \ge 3$, $x_3 \ge 4$, $x_4 \ge 5$?
(c) $2 \le x_1 \le 5$?
(d) $2 \le x_1 \le 5$ and $3 \le x_2 \le 7$?