There are n ≥ 4 points on a circle. Each pair of them is connected by a chord such that no three of these chords intersect at the same point inside the circle. What is the number of regions formed inside the circle?
I am a little confused on how to even begin with this problem. How exactly am I supposed to find where and how many intersections there are between the chords? This will allow me to split up the circle into regions. Along with this, how exactly will we find the number of vertices. I will need this information to apply Euler's formula and get the number of faces. But I am a little confused on how to do that.
Any help?
This is OEIS $A000127$: $$a_n={n\choose 4}+{n\choose 2}+1={n^4 - 6n^3 + 23n^2 - 18n + 24\over24}$$
It starts surprisingly by powers of $2$ ($1$, $2$, $4$, $8$, $16$), but this is treacherous.
I particularly like the proofs given here, one of which relies precisely on Euler's formula (you can count the number of vertices and edges of the picture, and hence obtain the desired number of faces).