Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle formed by these three vertices is equilateral is

If $p$ points are taken on each of the three coplanar parallel lines,what is the maximum number of triangles that can be formed with vertices at these points?

In how many ways can a mixed doubles tennis game be arranged from $7$ married couples if no husband and wife are in the same game?

The straight lines $l_1, l_2, l_3$ are parallel and lie in the same plane. $A$ total number of $m$ points are taken on $l_1$,$n$ points on $l_2$,and $k$ points on $l_3$. The maximum number of triangles formed with vertices at these points is:

The number of rectangles that can be obtained by joining four of the twelve vertices of a $12$-sided regular polygon is

There are $m$ points on a straight line $AB$ and $n$ points on another line $AC$,none of them being the point $A$. Triangles are formed from these points as vertices when $(i)$ $A$ is excluded $(ii)$ $A$ is included. Then the ratio of the number of triangles in the two cases is