I found that a circle placed in a rectangle can be shaded in 4 different ways. First, don't shade anything. Second, shading the circle only. Third, shading the rectangle only. Fourth, shading both the circle and rectangle.
The question is: Find the number of shadings possible for 2 circles in a rectangle in both situations: they don't overlap and overlap.
I found that 2 circles in a rectangle can be shaded in 24 different ways in both cases. I'd like to ask that can I use counting technique to solve this problem? If I can, can you show me how to use counting technique to solve this? Thank you very much.

For the case of two circles in a rectangle with no overlap between the circles - I count only 8 ways to shade them, not 24. Please show how you got 24.