A while ago I posted a question: Coloring a Grid.
Online, I seem to have stumbled upon a usage of PIE AOPS Wiki Solution AIME II #9.
(1) Now, I have experience with PIE, but I do not see how to divide sets in this context?
2026-03-31 07:52:18.1774943538
How to use Principle of Inclusion-Exclusion here?
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I understand your question to be how the principle of inclusion-exclusion is being used to determine the number of admissible colourings for a given number $n$ of pieces on the Art of Problem Solving page that you linked to.
The number of all possible colourings is $3^n$. From these, we need to subtract the ones that use only $2$ colours. We have $3$ choices of a colour to leave out, and then $2^n$ choices of colours for the pieces. But now we've subtracted the colourings that use only one colour twice instead of just once, so we need to add them back in once, and there are $3$ of them.
So that makes $3^n-3\cdot2^n+3$.