The number of diagonals in a polygon is $20$. The number of sides of the polygon is:

There are $10$ points on the line segment $AB$ excluding $A$ and $B$,and $8$ points on the line segment $AC$ excluding $A$ and $C$. The number of triangles formed using these $18$ points (excluding $A, B,$ and $C$) is:

In a plane,there are $10$ points,out of which $4$ are collinear. The number of triangles that can be formed by joining these points is:

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:

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 were two women participating in a chess tournament. Every participant played two games with every other participant. The number of games that the men played between themselves exceeded the number of games that the men played with the women by $66$. The total number of participants is