Counting points of intersection There are 9 points on the circumference of a circle. The points are not evenly spaced. 
Line segments are drawn connecting each pair of points.
What is the largest number of different points inside the circle at which at least two of these line segments intersect?
Here is what I am attempted trying to solve this problem. With 3 points on the circumference there will not any point of intersection. With 4th point I have 1 point of intersection. With 5 points I am getting 5 points of intersection (shape looking like a star with one spoke missing). 6th point make the total go up to 15. Do I need to keep counting this way or there is better method? Also I am failed to understand how evenly spaced or unevenly spaced points would have made a difference.

Every pair of points divides the set of remaining points into two sets (let say $L$ and $R$ for points left of the segment and right of the segment). Then, the segment formed by these two points will cross internally the segments with one end in $R$ and one end in $L$. There are exactly $|L| \cdot |R|$ such segments.
The size of $L$ is something between 0 and $n-2$. 
A segment for which $|L| = 0$ will have no internal crossing. There are $n$ such segments.
A segment for which $|L| = k$ will have $k(n-2-k)$ crossings. There are $n$ such segments.
In the end each crossing is counted twice so you are looking for
$$ \frac{n}{2}\sum_{k=0}^{n-2} k(n-2-k).$$
