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

If in a regular polygon the number of diagonals is $54$,then the number of sides of this polygon is

$15$ persons are sitting around a circular table. The number of ways of selecting three persons at a time from them,such that the selected three do not sit together at one place 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?

The number of diagonals in a regular polygon of $100$ sides is

Let $P_1, P_2, \ldots, P_{15}$ be $15$ points on a circle. The number of distinct triangles formed by points $P_i, P_j, P_k$ such that $i+j+k \neq 15$ is