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Two biologists, two chemists and two physicists sit at a round table with 6 chairs. In how many ways can they sit so that no two scientists of the same type are seated next to each other?

I don't understand the given PIE (Principle of Inclusion Exclusion) solution.

According to the book,

# of seatings with no pair adjacent = # of seatings (with no restriction) - # of seatings with 1 pair adjacent + " 2 pair adjacent - " 3 pairs adjacent

I don't understand why 2 pair adjacent is being added to the problem. If we eliminate with all seatings with 1 pair adjacent, does that not mean we also eliminate all invalid options? Isn't it true that there cannot be a valid option/arrangement in the set of (1 pair adjacent)--no matter how you arrange the other 4 scientists once you have 1 pair adjacent in the set--everything is invalid. I see how the solution makes sense in other examples, but the idea of a circular table is throwing me off tremendously. I can't seem to create concrete examples of repeats, when I try to draw out the tables and subsequently position the people.

Could someone please explain the second step (1 pair and 2 pair)? Thanks.