GATE 2024 CE (CE1) – 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 of twins must sit on two adjacent chairs, so treat each pair as one block of 2 chairs. The twins in a pair are indistinguishable, so a block has no internal order. The table then has 3 different blocks (6 chairs) and 2 empty chairs, which are identical. Arranging 5 objects around a circle, with 2 of them identical, gives $\frac{(5 - 1)!}{2!} = \frac{24}{2} = 12$.