Question

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?

• A. 12
• B. 14
• C. 10
• D. 28

Hint:

In circular arrangement problems with indistinguishable items, first group identical individuals (like twins into a block), then apply the circular permutation formula $(n-1)!$, and finally divide by factorials of identical groups (such as empty chairs).

Answer 1

The Correct Option is A

Step 1: Represent the problem. We have 3 sets of twins $\{A,A\},\{B,B\},\{C,C\}$ and $2$ empty chairs. The twins in each pair must sit together, so each pair behaves like a "block." Thus we have: - 3 twin blocks ($AA, BB, CC$), - 2 empty chairs. So, total $5$ objects to arrange around a circular table.

Step 2: Circular arrangements. For $n$ distinct objects around a circle, the number of arrangements is $(n-1)!$. Here: \[ (5-1)! = 4! = 24. \] 

Step 3: Adjust for indistinguishability within pairs. Within each twin pair, the order doesn't matter (since twins are indistinguishable). Thus, no further division is needed because each pair is already treated as a block. However, we must also note that the two empty chairs are indistinguishable. So we divide by $2!$: \[ \frac{24}{2} = 12. \] 

Step 4: Final Answer. Therefore, the number of unique seating arrangements is: 

\[ \boxed{12} \]