Question

In how many ways can five girls and five boys in a choir stand in a line if boys and girls must alternate positions?

Answer 1

To alternate positions, girls or a boy must occupy the first position on the line. The line will be GBGBGBGBGB, or BGBGBGBGBG where G represent girls, while B represent boys.

Below is the number of ways of selecting occupants of 1st to 9th positions for the two lines.

• 1, 5 ways
• 2, 5 ways
• 3, 4 ways
• 4, 4 ways
• 5, 3 ways
• 6, 3 ways
• 7, 2 ways
• 8, 2 way
• 9, 1 way
• 10, 1 way

Thus, the number of ways that the participants can form a line is $5^2×4^2×3^2×2^2×1^2=25×16×9×4=14400$

Since the two possible lines have possible ways of selecting a participant at any given positions, the total number of possible ways of forming a line is $14400×2=28800$