Finding the greatest number of regions on a plane 
There are 3 straight lines and two circles on a plane. They divide the plane into regions. Find the greatest possible number of regions.


This is the best that I could come up with an I don't really know how to get even more regions. Using some mental calculations, I figured that I could edit the picture and make it into 20 regions. However, the answer key gave me 21 as the answer. Is there a way to find out the number of regions on a plane without using a calculator? Is there a formula for it? Please help.

The number $21$ for a general configuration with maximal number of intersections follows from Euler’s formula. There are $18$ vertices ($3$ from line/line, $12$ from line/circle, $2$ from circle/circle intersections, and $1$ at infinity). There are $37$ edges ($21$ from lines, $16$ from circles). So $18 - 37 + A=2$ and $A=21$.
