If $x$ is the number of ways in which six women and six men can be arranged to sit in a row such that no two women are together and if $y$ is the number of ways they are seated around a table in the same manner,then $x: y=$ (in $: 1$)

The number of ways in which $3$ boys and $2$ girls can sit on a bench so that no two boys are adjacent is . . . . . .

The number of triangles that can be formed by $5$ points on a line and $3$ points on a parallel line is

Let $n > 2$ be an integer. Suppose that there are $n$ Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track. Further,each pair of nearest stations is connected by a blue line,whereas all remaining pairs of stations are connected by a red line. If the number of red lines is $99$ times the number of blue lines,then the value of $n$ is:

If three vertices are chosen from the six vertices of a regular hexagon and joined to form a triangle,what is the probability that the triangle formed is an equilateral triangle?

$p$ points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is