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 ( p / m + n ).

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 ( p / m + n ). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

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 × 4 ! × 5 C 2 × 3 ! × 2 !=4 ⋅ 4 ! 5 ! m + n =(1+4) ⋅ 4 ! 5 !=5 ! 5 ! text and p =5 ! 5 ! 2 ! ∴ ( p / m + n )=2

m= number of ways in which girls are consecutive =4!5! n= number of ways in which exactly 3 boys are consecutive =4C3​×4!×5C2​×3!×2!=4⋅4!5! m+n=(1+4)⋅4!5!=5!5! and p=5!5!2! ∴m+np​=2

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