In how many different ways can 6 girls and 6 boys form a circle such that the…

Concept: For n men and n women to be seated alternately around a circle, first fix one person from one group to remove the circular rotational symmetry; the remaining (n minus 1) members of that same group can then be arranged in (n minus 1) factorial ways. This automatically creates n alternating gaps for the other group, which can be filled in n factorial ways. So the total number of alternating circular arrangements equals (n minus 1) factorial multiplied by n factorial.

Applying this to the problem:

1. Fix one girl's position to remove the circular symmetry, since a circular arrangement has no fixed starting point.

2. The remaining 5 girls can now occupy the other girl-positions in 5 factorial = 120 ways.

3. Since boys and girls must alternate, fixing the 6 girls' seats creates exactly 6 distinct gaps between consecutive girls for the 6 boys.

4. These 6 boys can be arranged in the 6 gaps in 6 factorial = 720 ways.

5. Total number of arrangements = 5 factorial multiplied by 6 factorial = 120 multiplied by 720 = 86400.

Cross-check: Cross-check with a smaller case: for 2 girls and 2 boys alternating around a circle, the same rule gives 1 factorial multiplied by 2 factorial = 2 arrangements. Listing them directly -- fix one girl, the other girl takes the single remaining girl-seat, and the two boys can occupy the two gaps in 2 ways -- indeed gives exactly 2 distinct seatings, confirming the rule used above.

So the required number of ways = 86400.