Geometry and Combinatorics There are $p \gt 5$ points on the circumference of a circle, and every two of them are joined by a chord. Assume that no three chords are concurrent within the interior of the circle. How can I find the number of line segments obtained by dividing the chords through their points of intersections?
 A: Maybe someone is asking what would one do with the the numbers of line segments that jvdhooft correctly calculated in his answer. It is used to calculate number of faces chords divide the circle into. To do that we need to observe the polygon and the circle as one graph with $p + {p\choose4}$ vertices and $p + {p \choose 2} + 2{p \choose 4}$ edges (circle segments count). I'll elaborate more about this now. To calculate the number of faces we need to use Eulers graph formula, but it can only be used on graphs with non-intersecting edges. So our vetices need to be the starting $p$ points plus the number of intersections $p\choose4$ and our edges need to be the number of all line segments plus the number of circle segments. To calculate the number of line segments we can think about smaller number of points and lines.
In this picture there are three lines intersecting at two points. So we can see that there are $7 = 3 + 2*2$. In a general case numbers of line segments will be the number of lines plus  two times the number of intersections. Another way to think about this is like jvdhooft said. Now we can apply the Eulers formula: $$F = E - V + 2 = {p\choose2} + {p\choose4} +2$$
where $F$ is number of faces $V$ number of vertices and $E$ number of edges of the graph For more information see: https://www.youtube.com/watch?v=K8P8uFahAgc&t=417s
