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I have been very obsessed with a board game called Hex lately. I am trying to calculate the number of completed states (all positions filled in) on a 3x3 board where a certain player wins, given that some positions are already filled. I have attached a diagram illustrating what I am trying to do.

In the diagram are illustrations of a 3x3 Hex board, where pieces are played on the intersections. Each intersection either has a white or black piece on it, or is empty, meaning that it does not matter what color is placed on it.

For set A, the image designates 5 empty intersections that can either be black or white, giving a total of 2^5 elements in the set. Set B is the relative complement of another board with set A, which is equivalent to a set shown that includes 2^4 elements.

My problem is that I am having trouble calculating the number of elements in set C, which is equivalent to the relative complement of the board pictured with the union of sets A and B. It is not equivalent to any single diagram I can think of, so I can not compute it like I did with set B.

Are there any formulas I can use to compute this and further relative complements of unions?

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# Determining the number of elements in the relative complement of a set

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