The number of ways in which $6$ boys and $4$ girls can be arranged in a row such that between any two girls there must be exactly $2$ boys is

$6$ boys and $5$ girls sit in a line such that $(I)$ no two girls sit together $(II)$ all the girls sit together. If $p$ is the number of arrangements in case $(I)$ and $q$ is the number of arrangements in case $(II)$,then $p/q =$

In how many ways can $5$ girls and $3$ boys be seated in a row so that no two boys are together?

The number of integers greater than a million that can be formed using the digits $2, 3, 0, 3, 4, 2, 3$ is

In how many ways can $6$ people be arranged in a row such that $A, B, C,$ and $D$ always appear in the order $ABCD$ (not necessarily together)?

There are $10$ lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is