In how many ways can 5 boys and 5 girls sit around a circular table so that boys and girls alternate in the seating?

Introduction / Context:We require strict alternation of genders around a circle. Seat one gender first to create fixed slots for the other gender.

Given Data / Assumptions:

• 5 distinct boys, 5 distinct girls.
• Single round table; rotations identified (circular permutations).
• Alternation constraint: B–G–B–G–B–G–B–G–B–G.

Concept / Approach:Arrange boys in a circle first: (5−1)! = 4!. Then place the 5 girls into the 5 gaps between boys: 5! ways.

Step-by-Step Solution:Arrange boys: 4! = 24.Place girls in gaps: 5! = 120.Total = 24 * 120 = 2880.

Verification / Alternative check:Starting with girls gives 4! ways similarly and then 5! for boys—same product. Alternation enforces unique interleaving slots.

Why Other Options Are Wrong:Nearby values (2800, 2680, 2280) are arithmetic slips or stem from dividing by an extra symmetry that does not apply.

Common Pitfalls:Using 5! for the first gender (forgetting circular reduction) or multiplying by 2 for reversing direction (unnecessary for standard circular seating).

Final Answer:2880