If 10 boys and 10 girls sit alternately in a row and then sit alternately along a circular table, what is the ratio of the number of ways of sitting in a row to the number of ways of sitting along a circle?
Hint 1: Calculate the number of row arrangements: it is since we can start with either a boy or a girl.
Hint 2: Calculate circular arrangements: fixing one boy, the boys can sit in ways, and the girls in ways.
Hint 3: Divide the row ways by circular ways: .
Step 1 (Calculate row seating arrangements): We want to sit 10 boys and 10 girls alternately in a row:
Case 1: Starting with a boy (B G B G ...). There are ways to arrange the boys and ways to arrange the girls: .
Case 2: Starting with a girl (G B G B ...). Similarly, there are ways.
Total row ways:
Step 2 (Calculate circular table seating arrangements): In a circle, we fix one person to break rotation symmetry:
Fix one boy's position. The remaining 9 boys can be seated in ways.
Since seating is alternate, the 10 girls must sit in the 10 spaces between the boys, which can be arranged in ways.
Total circular ways:
Step 3 (Compute the ratio): We divide the two values:
Step 4 (Conclusion): The ratio of the seating arrangements is exactly 20.
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