Discussion: After doing a few cases and, in particular, analyzing closely the case of exactly four points, this student realizes that for each choice of four points there is exactly one choice of a pair of chords of the circle which give an intersection point and therefore in the general case the number of intersections is n choose 4.

Written response; Good investigation and pattern recognition. You got to the cruz of the problem by seein that every choice of four points gives rise to one intersection and therefore the number or such intersections is "n choose 4."