Please suggest different ways to solve this question

# Doubt from Permutations and Combination

Find the total number of possibilities of colouring:

2^9 = 512.

Subtract 2x2 red square possibilities - 4.

That gives 508.

Not sure how u get 417

This uses Principle of Inclusion-Exclusion. You will see that this will be similar to n(A \bigcup B \bigcup C \bigcup D)

There are four 2 \times 2 squares in the figure, name them A,B,C,D in clockwise order.

There are a total of 2^9=512 colourings possible. From these

**Subtract:** Case I: One of the squares is painted fully red. The remaining 5 squares can be filled in 2^5 ways. Thus we have 4\times 2^5 = 128 such colourings

**Add**: The above include Case II: Where two sets of squares are painted red. Here we have to distinguish adjacent pairs (A,B), (C,D), (A,D), (B,C) which gives us 4 \times 2^3 = 32 colourings, and diagonal pairs A,C and B,D which gives 2 \times 2^2=8 colurings.

Hence from this case we have a total of 40 colourings

**Subtract**: Case III: Three sets of squares are filled in red. This gives 4 \times 2 =8 colourings.

**Add**: Case IV: All four sets are filled in red: 1 colouring

Hence number of permissible colurings = 512-128+40-8+1=417

Sir, is their is any method for doing this question.

@Hari_Shankar @Praveen_2018

Can't we count no of red colour 2*2 square and then substract it.

There will be too much overlapping of cases.