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

There are $m$ men and $2$ women participating in a chess tournament. Each participant plays $2$ games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84$,then the value of $m$ is

$A$ set of $10$ parallel lines is intersected by another set of $m$ parallel lines. If the total number of parallelograms formed by these two sets of lines is $675$,then $m$ is equal to:

The number of parallelograms that can be formed from a set of $4$ parallel lines intersecting another set of $3$ parallel lines is:

$A$ polygon has $54$ diagonals. Then,the number of its sides is

There are $6$ points on a circle. Two triangles are drawn such that they have no common vertices. What is the probability that none of the sides of the triangles intersect?