Question

Let there be 4 boys and 4 girls are standing in a row. If $m$ is the number of ways in which all the girls are consecutive and $n$ is the number of ways in which exactly 3 boys are consecutive. If $p$ denotes the number of ways in which 5 gentlemen \& 5 ladies can stand in a row such that ladies and gentlemen are alternate then find the value of $\frac{ p }{ m + n }$.

Answer 1

$m =$ number of ways in which girls are consecutive $=4 ! 5 !$
$n =$ number of ways in which exactly 3 boys are consecutive
$={ }^4 C _3 \times 4 ! \times{ }^5 C _2 \times 3 ! \times 2 !=4 \cdot 4 ! 5 !$
$m + n =(1+4) \cdot 4 ! 5 !=5 ! 5 ! \text { and } p =5 ! 5 ! 2 ! $
$\therefore \frac{ p }{ m + n }=2$