If a five-digit number divisible by $3$ is to be formed using the digits $0, 1, 2, 3, 4,$ and $5$ without repetition,then the total number of ways this can be done is:

The number of positive integral solutions of the equation $xyz = 3000$ is

The total number of integral solutions of the equation $xyz = 24$ is

Suppose $a_2, a_3, a_4, a_5, a_6, a_7$ are integers such that $\frac{5}{7} = \frac{a_2}{2!} + \frac{a_3}{3!} + \frac{a_4}{4!} + \frac{a_5}{5!} + \frac{a_6}{6!} + \frac{a_7}{7!}$,where $0 \leq a_j < j$ for $j = 2, 3, 4, 5, 6, 7$. The sum $a_2 + a_3 + a_4 + a_5 + a_6 + a_7$ is:

How many words can be formed using $3$ consonants and $2$ vowels from $5$ consonants and $4$ vowels?

Find the remainder when $(1!)^2 + (2!)^2 + (3!)^2 + \dots + (100!)^2$ is divided by $10^2$.