Question

In how many ways can 6 boys and 5 girls be arranged for a group photograph if the girls are to sit on chairs in a row and the boys are to stand in a row behind them?

Answer 1

Given : Number of boys = 6 and number of girls = 5 

To find : Possible number of arrangements in a group photograph 

Let boys be b₁, b₂, b₃, b₄, b₅, b₆ and girls be g₁, g₂, g₃, g₄, g₅ 

Possible arrangements are :

b₁ b₂ b₃ b₅ b₆ b₄ 

g₂ g₄ g₁ g₅ g₃ 

b₂ b₁ b₅ b₃ b₄ b₆ 

g₂ g₄ g₅ g₁ g₃ 

In this arrangement, 

We are arranging boys and girls separately. 

Formula used : 

Number of arrangements of n things taken all at a point = P(n, n)

P(n, r) = \(\frac{n!}{(n-r)!}\)

∴ Number of ways to arrange boys,

= the number of arrangements of 6 things taken all at a time 

= P(6, 6)

= \(\frac{6!}{(6-6)!}\)

= \(\frac{6!}{0!}\)

{∵ 0! = 1} 

= 6! 

= 6 × 5 × 4 × 3 × 2 × 1 

= 720

Formula used : 

Number of arrangements of n things taken all at a point = P(n, n)

P(n, r) = \(\frac{n!}{(n-r)!}\)

∴ Number of ways to arrange girls,

= the number of arrangements of 5 things taken all at a time.

= P(5, 5)

= \(\frac{5!}{(5-5)!}\)

= \(\frac{5!}{0!}\)

{∵ 0! = 1} 

= 5! 

= 5 × 4 × 3 × 2 × 1 

= 120 

Now, 

We will get total number of ways by multiplying their separate arrangements 

∴ Total number of ways = 720 × 120 

= 86400 

Hence,

Possible number of arrangements in which 6 boys and 5 girls can be arranged for a group photograph with provided conditions are 86400.