Find the number of marked points on the plane,if when connected pairwise by line segments,the total number of line segments formed is $15$.

$10$ distinct points are taken on a circle. Then using these points:
Statement $I$: The number of triangles that can be formed is $100$
Statement $II$: The number of chords that can be formed is $45$
Which of the following is correct?

Let $n > 2$ be an integer. Suppose that there are $n$ Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track. Further,each pair of nearest stations is connected by a blue line,whereas all remaining pairs of stations are connected by a red line. If the number of red lines is $99$ times the number of blue lines,then the value of $n$ 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

If three vertices are chosen from the six vertices of a regular hexagon and joined to form a triangle,what is the probability that the triangle formed is an equilateral triangle?

The number of diagonals in an octagon is