Suppose that $20$ pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars,then the total number of beams is

In a plane,there are $37$ straight lines,of which $13$ pass through point $A$ and $11$ pass through point $B$. Moreover,no three lines (apart from the lines passing through $A$ and $B$) pass through the same point,and no two lines are parallel. What is the number of points of intersection of the straight lines?

There are $12$ points in a plane,no three of which are in the same straight line,except $5$ points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these $12$ points is

Let $T_n$ denote the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides. If $T_{n+1} - T_n = 21$,then $n$ equals

The number of shortest paths from point $A$ to $D$ (as shown in the figure) is:

Out of $30$ points in a plane,$8$ of them are collinear. The number of straight lines that can be formed by joining these points is: