Circular Permutations

Circular permutations of n distinct objects number (n - 1)! because rotations of the same arrangement are considered identical, a key JEE Advanced distinction from linear arrangements. With 8 people, fixing one person to remove rotational symmetry leaves 7 others to arrange in the remaining seats, giving (8 - 1)! = 7! = 5040. The reduction from 8! to 7! reflects that a circular table has no fixed starting point. Option 40320 = 8! counts linear arrangements, overcounting each circular arrangement 8 times. Option 720 = 6! uses the wrong reduction. Option 362880 = 9! is unrelated. Hence the number of seatings is 5040. Plausibility check: the ratio of linear to circular counts should equal the number of rotations, and 8!/7! = 8, exactly the number of equivalent rotations of a circle of 8 seats, confirming the (n - 1)! formula. Whether to use permutations or combinations hinges entirely on whether order matters, and committee or selection problems are the canonical setting where it does not. The independence of the two group selections justifies multiplying their counts, and this product structure recurs whenever a configuration is built from several unrelated unordered choices made simultaneously across distinct categories.