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 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

Let us calculate $n$ first.

We can consider $5$ girls as one set and the $5$ boys. We can arrange them in $6!$ ways. The girls in the set can be arranged in $5!$ ways.
Hence, $n=5!\times 6!$
Let us calculate $m$.

Now, let us first place the 4 girls together. We will need to consider different cases.

Case 1 : $4$ girls at the corner. $4$ girls can be selected and permuted in ${}^5{P}_4=5!$ ways. The $5^{th}$ girl can be placed in $5$ ways. The boys can be placed in $5!$ ways. They can be placed in the left or the right corner.

Cases $=2\times 5\times 5!\times 5!$
Case 2: The $4$ girls are not placed at the corner.
The position of the $4$ girls together can be selected in $5$ ways. The $4$ girls can be selected and permuted in ${}^5{P}_4$ ways. The $5^{th}$ girl can be placed in $4$ ways. The $5$ boys can be placed in $5!$ ways.

Cases $=5\times 5!\times 4\times 5!$

Hence, $m=30\times 5!\times 5!$
Hence, $\frac{m}{n}=5$