Question

Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number in which 5 boys and 5 girls stand in such a way that exactly four girls stand consecutively in the queue. Then the value of `m/n` is ____
A. 4
B. 5
C. 6
D. 7

Answer 1

All five girls can stand consecutively in a queue in 5! Ways. Considering these five girls as an individual and mixing up with 5 boys there are 6 individuals who can stand in a queue in 6! Ways. Therefore, `n=5!xx6!`
In order to find the number of ways in which exactly four girls stand consecutively. Let us first choose 4 girls out of 5. This can be done in `""^(5)C_(4)` ways. These 4 girls can stand in a queue in 4! ways. Now, consider these 4 girls as an individual and mix-up with remaining one girl and 5 boys. In this manner we obtain 7 persons which can be arrange in a row in 7! ways.
Thus.
number of ways in which four girls stand consecutively in a queue `""^(5)C_(4)xx4!xx7!`.
These ways also include the ways in which all five girls stand consecutively in a queue the number of such ways is `2(""^(5)C_(4)xx4!xx6!)`
`:.` Number of ways also include which exactly 4 girls stand is a queue is `""^(5)C_(4)xx4!xx7!-2(""^(5)C_(4)xx4!xx6)`.
i.e. `m=""^(5)C_(4)xx4!xx7!-2(""^(5)C_(4)xx4!xx6!)=""^(5)C_(4)xx4!xx6!xx5=5xx5!xx6!`
`:.(m)/(n)=(5xx5!xx6!)/(5!xx6!)=5`.