$15$ girls are seated at a round table. The number of ways of selecting three girls such that all the three are not seated together is

Let $p_n$ denote the total number of triangles formed by joining the vertices of an $n$-sided regular polygon. If $p_{n+1} - p_n = 66$,then the sum of all distinct prime divisors of $n$ 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 polygon is $20$. The number of sides of the polygon is:

$A$ parallelogram is cut by two sets of $m$ lines parallel to its sides. The number of parallelograms thus formed is

Statement-$1$: If a polygon has $45$ diagonals,then the number of sides is $10$. Statement-$2$: Out of $n$ non-collinear points,$2$ points can be selected in $^nC_2$ ways.