Inclusion exclusion principle

set up S It's a finite set ,a1,a2,……,an yes n Nature

remember N(ai) by S There is ai Number of elements of property . special , remember N(1)=|S|.

remember N(1-ai) by S There is no ai Number of elements of property .

N(aj1,aj2,……,ajk) by S At the same time aj1,aj2,……,ajk Number of elements of property

remember N(a+b)=N(a)+N(b)  N(a-b)=N(a)-N(b)

Then the principle of inclusion and exclusion can be written as ：

When it is not particularly easy to solve a problem directly, we can consider its opposite , Then we use the principle of inclusion and exclusion to solve , It often makes the problem simple .