Digit Bounded Numbers

You are given $N$ intervals $[L_1, R_1], [L_2, R_2], \ldots, [L_N, R_N]$. Consider $N$ integers $x_1, x_2, \ldots, x_N$ such that: - $x_i \in [L_i, R_i]$ for each valid $i$ - all decimal digits of $S = x_1 + x_2 + x_3 + \ldots + x_N$ (without leading zeroes) lie between $A$ and $B$ inclusive In how many different ways can you pick the sequence $x_1, x_2, \ldots, x_N$? Since this number may be