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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?

1. 12
2. 14
3. 10
4. 28

Correct answer: 12

Solution

To find the unique seating arrangements, we can treat each pair of twins as a single unit or block. With three pairs, we have three blocks to arrange around the circular table, which can be done in (3-1)! = 2! = 2 ways. Each pair can be arranged internally in 2 ways, leading to a total of 2 * 2 * 2 = 8 arrangements. Thus, the total unique arrangements are 2 * 8 = 16, but since we need to account for indistinguishable twins, we divide by 2 for the pairs, resulting in 12 unique arrangements.

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