In how ways three girls and nine boys can be seated in two vans, each

1.	In how ways three girls and nine boys can be seated in two vans, each having numbered seats, 3 in the front and 4 at the back? How many seating arrangements are possible if 3 girls should sit together in a back row on adjacent seats?
Answer» We have 14 seats on two vans and there are 9 boys and 3 girls i.e total 12 peoples The number of ways of arranging 12 peoples on 14 seats without restriction is ¹⁴P₁₂ ¹⁴P₁₂ = \(\frac{14!}{2!}\) = \(\frac{17∙13!}{2!}\) = 7 × 13! Ways. Three girls can be seated together in back row on adjacent seats in the following ways 1,2,3, or 2,3,4 of first van 1,2,3 or 2,3,4 of second van In each way the three girls can interchange among themselves in 3! Ways ∴ total number of ways in which 3 girls 3 girls sit together in a back row = 4 × 3! = 24 ways 9 boys are to seated on 11 seats = 11P9 = \(\frac{11!}{2!}\) Hence, the total number of ways \(\frac{24\times11}{2!}\) = 12 × 11!

In how ways three girls and nine boys can be seated in two vans, each having numbered seats, 3 in the front and 4 at the back? How many seating arrangements are possible if 3 girls should sit together in a back row on adjacent seats?

Answer»

We have 14 seats on two vans and there are 9 boys and 3 girls

i.e total 12 peoples

The number of ways of arranging 12 peoples on 14 seats without restriction is ¹⁴P₁₂

¹⁴P₁₂ = \(\frac{14!}{2!}\) = \(\frac{17∙13!}{2!}\)

= 7 × 13! Ways.

Three girls can be seated together in back row on adjacent seats in the following ways

1,2,3, or 2,3,4 of first van

1,2,3 or 2,3,4 of second van

In each way the three girls can interchange among themselves in 3! Ways

∴ total number of ways in which 3 girls 3 girls sit together in a back row = 4 × 3! = 24 ways

9 boys are to seated on 11 seats = 11P9 = \(\frac{11!}{2!}\)

Hence, the total number of ways \(\frac{24\times11}{2!}\) = 12 × 11!