Question

Let n be the number of ways in which $5$ boys and $5$ girls an stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which $5$ boys and $5$ girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue.Then the value of $\frac{m}{n}$ is.

Answer 1

• You can arrange
  

55 girls in a queue consecutively in 5!5! ways. Arrange the 55 boys in a queue in 5!5! ways. Insert the 5 girls anywhere in the queue of boys, there are 66 positions to choose from. This gives a total of 6×5!×5!6×5!×5! ways to do it.

• n= 6×5!×5!
  

• Consider
  

44 girls standing in a queue. To do that, choose 44 girls out of 55, and rearrange them in 4!4! ways - that is 5!5! combinations. Insert the queue of 44 girls within the queue of 55 boys (arranged in 5!5! ways among themselves) in 66 ways, as before. Now, there are 6 objects in the queue: each of the 5 boys, and the queue of 44 girls. The fifth girl can go anywhere except immediately before or after the queue of 44 girls - so, there're 55 possible places she can occupy. This gives us a total of 5!×5!×6×55!×5!×6×5 choices.

• m= 5!×5!×6×5
• m/n = (5!×5!×6×5)/ (6×5!×5!) = 5 ANSWER

5!×5!×6×5