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

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

How many necklaces can be formed by selecting $4$ beads from $6$ different colored glass beads and $4$ beads from $5$ different colored metal beads?

The greatest possible number of points of intersection of $8$ distinct straight lines and $4$ distinct circles is

Line $L_1$ with slope $2$ and line $L_2$ with slope $\frac{1}{2}$ intersect at the origin $O$. In the first quadrant,$P_1, P_2, \ldots, P_{12}$ are $12$ points on line $L_1$ and $Q_1, Q_2, \ldots, Q_9$ are $9$ points on line $L_2$. The total number of triangles that can be formed having vertices at three of the $22$ points $(O, P_1, P_2, \ldots, P_{12}, Q_1, Q_2, \ldots, Q_9)$ is:

The number of diagonals of a polygon is $35$. If $A$ and $B$ are two distinct vertices of this polygon,then the number of all those triangles formed by joining three vertices of the polygon having $AB$ as one of its sides is: