In how many ways can the four walls of a room be painted with three colours so that no two adjacent walls have the same colour ?
I specifically want to use inclusion exclusion principle. So according to me it should be $$3^4-4\times 3^3+4\times 3^2-3 $$ which is $6$. The correct answer is $18$. Am making a mistake in applying the principle ? I thought like this - Total ways to colour - ways to colour two adjacent walls with same colour + ways to colour three adjacent walls with same colour - ways to colour all four walls with same colour.