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: 

Number of boys = 5

Number of girls = 5

Number of boys can be seated in the row 5p5 = 5!

Number of girls can be seated in the row 6p5 = 6!

Number of ways in which no two girls sit together = 5! ✕ 6!

5 girls and 5 boys can arrange themselves in 2! ways.

5 girls can arrange among themselves in 5! ways.

5 boys can arrange among themselves in 5! ways.

Total number of ways of seating arrangements = 2! × 5! × 5!

= 2! × (5!)2

 

Total number of ways in which no girls are together = Total number of arrangements – Number of arrangements in which all the girls are together.

5 boys and 5 girls total number of arrangements of = 10P10 = 10! Ways.

∴ Total number of ways in which all the never girls sit together = 10! – 5! × 6!

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