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, use \( (n-1)! \) to account for rotational symmetry. For groups of identical items, multiply by the internal permutations within each group.

Answer 1

The Correct Option is A

\includegraphics[width=0.5\linewidth]{a7 CE.PNG} Step 1: Problem Setup. We are tasked with finding the number of unique circular arrangements of 5 units, out of which 2 are alike (\( E \) and \( E \)). Step 2: Formula for circular arrangements. The total number of arrangements in a circle, accounting for repetition, is given by: \[ \frac{(n-1)!}{k!}, \] where \( n \) is the total number of units and \( k \) is the number of identical units. Step 3: Substituting the values. Here, \( n = 5 \) and \( k = 2 \) (for \( E \) and \( E \)): \[ \text{Number of unique arrangements} = \frac{(5-1)!}{2!}. \] Step 4: Simplifying the factorials. \[ \text{Number of unique arrangements} = \frac{4!}{2!} = \frac{4 \times 3 \times 2 \times 1}{2 \times 1}. \] Step 5: Calculating the result. \[ \text{Number of unique arrangements} = 12. \]