The largest number of intersection points we can obtain from $10$ distinct points on a circle. Suppose we pick $10$ distinct points on a circle and connect all pairs of points with line segments, what is the largest number of intersection points we can obtain?

I have tried with $4$ points and got $1$ intersection point.
Next, I have tried with $5$ points and got $5$ intersection points.
Finally, I have tried with $6$ points and got $15$ intersection points.

After that I am lost. Can someone give an argument how to find it?

Pick any arrangement of the $n$ points that you like, choose $4$ points out of these $n$ points, call them $A,B,C,D$ and lets say that they have a circular order so if you start, say, on the top of the circle going clockwise you hit the points in lexicographic order. Is it true that they form exactly one intersecting point? If so, then the problem boils down to finding how many $4$ points you can choose out of $n$ points, so $\binom{n}{4}.$ This if you can guarantee that you could have find an arrangement in such a way that two intersecting points are not the same intersecting point.
