There are $k$ points placed on a circle. We draw all the segments between pairs of points. How many intersections can there be between these segments?
For each group of 4 vertices, we draw the 6 segments. This forms a quadrilateral with diagonals that intersect. We deduce that from 4 vertices, this gives 1 intersection. If there are $k$ points around the circle, this then gives $\binom{k}{4}$ intersections.