Three vertices are chosen randomly from the seven vertices of a regular $7$-sided polygon. The probability that they form the vertices of an isosceles triangle is

Out of $18$ points in a plane,no three are in the same straight line except five points which are collinear. The number of $(i)$ straight lines and $(ii)$ triangles which can be formed by joining them is:

Let $l_1$ and $l_2$ be two lines intersecting at $P$. If $A_1, B_1, C_1$ are points on $l_1$,and $A_2, B_2, C_2, D_2, E_2$ are points on $l_2$,and if none of these points coincides with $P$,then the number of triangles formed by these eight points is:

There are two women participating with some men in a chess tournament. Each participant played two games with every other participant. The number of games that the men played between themselves is $66$ more than the number of games that the men played with the women. Find the total number of participants in the tournament.

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle formed by these three vertices is equilateral is

The number of triangles whose vertices are the points $(x, y)$ in the $XY$-plane with integer coordinates satisfying $0 \leq x \leq 4$ and $0 \leq y \leq 4$ is: