Chord Intersections

From an intersection point, trace back along the chords to find four endpoints on the circle. Also, any such group of 4 (out of $n$ ) uniquely determines an intersection point. So there is a one-to-one correspondence between points of intersection and (unordered) groups of 4 points. The total number of such groups is the binomial coefficient $\binom{n}{4}$ , or $\frac{n(n-1)(n-2)(n-3)}{24}$ .

(In the example, where $n=7$ , there are $\binom{7}{4}=\frac{7\times 6\times 5\times 4}{4!} = 35$ intersections.)