Question

Find the number of ways in which 5 boys and 5 girls may be seated in a row so that no two girls and no two boys are together.

Answer 1

Number of boys = 5

Number of girls = 5

(2)

Two groups of 5 girls and 5 boys can be arranged in 2! ways.

5 girls can arrange among themselves in 5! ways.

5 boys can arrange among themselves in 5! ways.

Hence, total number of ways of seating arrangements = 2! × 5! × 5! = 2! × (5!)2

(3)

Total number of ways in which all the girls are never together = Total number of arrangements – Number of arrangement in which all the girls are always together.

Total number of arrangement of 5 boys and 5 girls = 10P10 = 10! ways.

∴Total number of ways in which all the girls are never together = 10! – 5! 6! (Using (1))