GATE 2024 XH2 (English) – Question 7
Three distinct sets of indistinguishable twins are to be seated at a circular table that has 8 identical chairs. Unique seating arrangements are defined by the relative positions of the people.
How many unique seating arrangements are possible such that each person is sitting next to their twin?
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Correct answer: (A) 12
Explanation
Each pair takes two adjacent chairs, so on the 8 chairs we place 3 pairs (blocks of 2) and 2 empty chairs. Tilings of a labelled 8-cycle with 3 blocks and 2 empty chairs number $\frac{8}{5}\binom{5}{3} = 16$. The three pairs are distinct, so there are $16 \times 3! = 96$ placements. Rotations of the table give the same arrangement (8 rotations, each giving a different placement), so there are $96 / 8 = 12$ arrangements.