Question

Let $n$ be the number of ways in 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 $\dfrac{m}{n}$, is ?

Answer 1

Hint: Use permutations and combinations to determine the number of possible ways in which girls and boys can stand in the queue according to the situations given in the question. Then, take the ratio of the number of arrangements obtained in the two different situations.

Complete step-by-step answer:

Our first situation is that all the girls are standing consecutively in the queue. Let us take the arrangement of boys first. There are 5 boys.
$\_{{B}_{1}}\_{{B}_{2}}\_{{B}_{3}}\_{{B}_{4}}\_{{B}_{5}}\_$
There can be six vacant places where 5 girls can stand consecutively. We have to select one place.
Total number of ways to select one place out of six places will be 6.
These 5 girls can be arranged among themselves in $5!$ ways.
Similarly, 5 boys can be arranged among themselves in $5!$ ways.
Therefore, $n=6\times 5!\times 5!$.
Now, consider the second situation where 4 girls are standing consecutively in the queue.
There will be 2 groups of girls. One group contains 4 girls and the other contains 1 girl. We have to select two positions to place the two groups.
Number of ways to select two places out of six places is ${}^{6}{{C}_{2}}$.
Number of ways to arrange two groups of girls is $2!$.

Number of ways to select a group of 4 girls among 5 girls will be ${}^{5}{{C}_{4}}$.
Now, 5 boys can be arranged among themselves in $5!$ ways and 4 girls can be arranged among themselves in $4!$ ways.
Therefore, \[m=2!\times 5!\times 4!\times {}^{6}{{C}_{2}}\times {}^{5}{{C}_{4}}\].
Now, \[\dfrac{m}{n}=\dfrac{2!\times 5!\times 4!\times {}^{6}{{C}_{2}}\times {}^{5}{{C}_{4}}}{6\times 5!\times 5!}=5\].

Note: We have to consider each way of arrangement properly like arrangement of boys and girls among themselves and ways to select 4 girls out of 5 who will stand consecutively. If any one of the arrangements is neglected or left, we will get the wrong answer.