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

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

Let $S = \{(a, b) : a, b \in \mathbb{Z}, 0 \leq a, b \leq 18\}$. The number of lines in $\mathbb{R}^2$ passing through $(0,0)$ and exactly one other point in $S$ is:

If $t_n$ denotes the number of triangles formed with $n$ points in a plane,no three of which are collinear,and if $t_{n+1}-t_n=36$,then $n$ is equal to

In a plane,there are $37$ straight lines,of which $13$ pass through the point $A$ and $11$ pass through the point $B$. Besides,no three lines pass through one point,no line passes through both points $A$ and $B$,and no two lines are parallel. The number of intersection points the lines have is equal to: