Question

Into how many parts can n circles divide the plane, maximum and minimum?

Answer 1

ALTERNATE APPROACH FOR MAXIMUM

It can be seen that there are 3 circles each one intersecting all the others.

There is one region that is not inside any circle (A).

There are three regions that are inside exactly one circle (B, C and D).

There are three regions that are inside exactly two circles (E, F and G).

There is one region that is inside all the circles (H).

Thus the total number of regions formed by three circles each one intersecting all the others is 1 + 3 + 3 + 1 = 8.

If there are n circles in a plane with each circle intersecting all the others, the number of regions formed would be as under:

Regions not inside any circle = C(n,0).

Regions inside exactly one circle = C(n,1).

Regions inside exactly two circles = C(n,2).

Regions inside exactly three circles = C(n,3).

Regions inside exactly r circles = n = C(n,r).

Regions inside exactly n circles = n = C(n,n).

So, the total number of regions into which the plane is divided is