In how many ways can 5 boys and 5 girls sit around a circle such that no two boys sit adjacent and no two girls sit adjacent?

Arrange the boys around the circle. Since the arrangement is circular, the number of ways to arrange n distinct objects in a circle is (n−1)!. So, the number of ways to arrange the 5 distinct boys around a circle is (5−1)!=4!=4×3×2×1=24. Place the girls in the spaces created by the boys. When the 5 boys are seated around the circle, they create 5 distinct spaces between them where the girls can be seated. _ B _ B _ B _ B _ B _ where B represents a boy and _ represents a space for a girl. Since no two girls can sit adjacent, each girl must occupy one of these 5 spaces. There are 5 distinct girls to be placed in these 5 distinct spaces. The number of ways to arrange these 5 girls in the 5 spaces is 5!=5×4×3×2×1=120. Step 3: Combine the arrangements of boys and girls. The total number of ways to arrange the 5 boys and 5 girls such that no two boys sit adjacent and no two girls sit adjacent is the product of the number of ways to arrange the boys and the number of ways to arrange the girls in the spaces between the boys. Total number of ways = (Number of ways to arrange boys) × (Number of ways to arrange girls) Total number of ways = 4!×5!=24×120=2880. Thus, there are 2880 ways for 5 boys and 5 girls to sit around a circle such that no two boys sit adjacent and no two girls sit adjacent.