Browsing the CTK I found your question, and after the clear explanation given by Stuart you probably got the number of regions as stated depends on the size of radius of the circumference. Another good example, is taking five points the pentagon area beyond the edges of the five points star can be or not within the circle.

However I think as an interesting alterative to your problem you can try to get the maximum number of regions that we can form taking the N points p1, p2.....pN within a circle (C), and connecting them with circumferences.
All circumferences will contain the center O of the original circle (C), and any pair of points: ( Op1p2; Op1p3; ...Op1pN,...Op2p3,Op2p4...Op(N-1)p(N) ).