In how many distinct ways can four people sit around a circular table, where two seatings are considered the same if one can be rotated to obtain the other?

Introduction / Context:
This question tests circular permutations. When people sit around a round table, rotating everyone together does not create a new arrangement, so we must adjust the usual linear permutation formula. Understanding this adjustment is crucial for many seating arrangement questions in aptitude tests.

Given Data / Assumptions:

There are 4 distinct people.

They are seated around a circular table.

Two arrangements are considered identical if one can be rotated to look like the other.

Concept / Approach:
For n distinct objects arranged in a circle, the number of distinct circular permutations is (n - 1)!. This formula accounts for the fact that rotating everyone together does not change the relative seating. Here n = 4, so we use (4 - 1)! = 3!.

Step-by-Step Solution:
Step 1: If the seats were in a straight line, the number of permutations would be 4! = 24.Step 2: On a circular table, each circular arrangement can be rotated into 4 different linear arrangements.Step 3: Therefore, we divide the linear count by 4 to remove this overcounting.Step 4: Number of circular arrangements = 4! / 4 = 24 / 4 = 6.Step 5: This is equal to (4 - 1)! = 3! = 6, which confirms the result.

Verification / Alternative check:
We can fix one person’s position as a reference, because the table is circular. If we imagine one person always sitting at the “top” position, the remaining 3 people can be arranged in 3! = 6 ways around them. This is another way to see that the answer is 6.

Why Other Options Are Wrong:
3 undercounts the possible orders and might come from a mistaken belief that only limited swaps are distinct.
9 does not correspond to any standard circular permutation formula for 4 distinct objects.
12 is half of 24 and would be correct only if we identified some, but not all, rotations or reflections, which is not the condition given in the question.

Common Pitfalls:
A common error is to forget the circular nature and simply answer 4!, which is 24. Another mistake is to divide by 2 instead of by 4, treating rotations and reflections together without clearly defining what counts as the same. Always remember: for circular permutations where only rotation is ignored and objects are distinct, the correct formula is (n - 1)!.

Final Answer:
The number of distinct circular arrangements is 6.