My answer is reference
Regions of a Circle Cut by Chords to n Points
---------------------------------------------- n points are distributed round the circumference of a circle and each point is
joined to every other point by a chord of the circle. Assuming that
no three chords intersect at a point inside the circle we require the
number of regions into which the circle is divided.
With no lines the circle has just one region. Now consider any
collection of lines. If you draw a new line across the circle which
does not cross any existing lines, then the effect is to increase the
number of regions by 1. In addition, every time a new line crosses an
existing line inside the circle the number of regions is increased by
1 again.
So in any such arrangement
number of regions = 1 + number of lines + number of interior
intersections
= 1 + C(n,2) + C(n,4)
Note that the number of lines is the number of ways 2 points can be
chosen from n points. Also, the number of interior intersections is
the number of quadrilaterals that can be formed from n points, since
each quadrilateral produces just 1 intersection where the diagonals
of the quadrilateral intersect.
Examples:
n=4 Number of regions = 1 + C(4,2) + C(4,4) = 8
n=5 Number of regions = 1 + C(5,2) + C(5,4) = 16
n=6 " " = 1 + C(6,2) + C(6,4) = 31
n=7 " " = 1 + C(7,2) + C(7,4) = 57