Ratio of Queue Permutations with Girls Standing Consecutively

To find the value of the ratio $\frac{m}{n}$, we analyze the two given conditions step by step.

Step 1: Calculate $n$

We are given that $n$ is the total number of arrangements of $5$ boys and $5$ girls in a queue such that all $5$ girls stand consecutively.

1. Treat the $5$ girls as a single combined block $G$.
2. We now have $5$ boys and $1$ block of girls, giving a total of $6$ items to arrange in a queue. These $6$ items can be arranged in $6!$ ways.
3. Within the block, the $5$ girls can be arranged among themselves in $5!$ ways.

Therefore, the total number of such arrangements is: $n = 6! \times 5!$

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Step 2: Calculate $m$

We are given that $m$ is the number of arrangements where exactly four girls stand consecutively together. This implies that four girls form one consecutive block, while the fifth girl is separated from this block by at least one boy (so that all five girls do not end up together).

1. Selection and Internal Arrangement of the Girls:

   • Choose $4$ girls out of $5$ to form the block $G_{(4)}$ in $\binom{5}{4} = 5$ ways.
   • The $4$ girls inside the block $G_{(4)}$ can be arranged among themselves in $4!$ ways.
   • The remaining girl $G_{(1)}$ can be viewed as an individual entity.

2. Arrangement of Boys and Placement of Girl Entities:

   • First, arrange the $5$ boys in a queue in $5!$ ways.
   • The $5$ boys create $6$ available gaps (including the two ends): $\text{\underline{\hspace{0.5cm}} } B_1 \text{ \underline{\hspace{0.5cm}} } B_2 \text{ \underline{\hspace{0.5cm}} } B_3 \text{ \underline{\hspace{0.5cm}} } B_4 \text{ \underline{\hspace{0.5cm}} } B_5 \text{ \underline{\hspace{0.5cm}}}$
   • To ensure that the block $G_{(4)}$ and the single girl $G_{(1)}$ are not adjacent, we must place them into two distinct gaps out of the $6$ available gaps.
   • The number of ways to choose $2$ distinct gaps and assign $G_{(4)}$ and $G_{(1)}$ to them is $^6P_2 = 6 \times 5 = 30$ ways.

Therefore, the total number of such arrangements is: $m = \binom{5}{4} \times 4! \times 5! \times {}^6P_2$ $m = 5 \times 4! \times 5! \times 30 = 5! \times 5! \times 30$

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Step 3: Compute the Ratio $\frac{m}{n}$

Now, taking the ratio of $m$ and $n$: $\frac{m}{n} = \frac{5! \times 5! \times 30}{6! \times 5!}$

Simplifying the fraction: $\frac{m}{n} = \frac{30 \times 5!}{6!} = \frac{30}{6} = 5$

Thus, the value of $\frac{m}{n}$ is $5$.