Question

# Six boys and four girls enter a railway compartment having 5 seats on each side. In how many ways can they occupy the seats if the girls are to occupy only the corner seats?

Hint- The total number of corner seats in a railway compartment are four, two on either side of the compartment.

6 boys and 4 girls enter a railway compartment having 5 seats on each side.
$\Rightarrow$Total seats$= 10$
$\Rightarrow$Total number of corner seats $= 4$(2 on either side)
$\Rightarrow$Remaining seats$= 10 - 4 = 6$
Hence 4 girls can occupy the 4 corner seats in 4! Ways.
Remaining 6 seats can be occupied by 6 boys in 6! Ways.
$\Rightarrow$Total number of ways for occupying the seats if girls occupy only corner seats
${\text{ = 4!}} \times {\text{6! = }}\left( {4 \times 3 \times 2 \times 1} \right)\left( {6 \times 5 \times 4 \times 3 \times 2 \times 1} \right) = 17280$
So, 17280 is the required total number of ways for occupying the seats if girls occupy only corner seats.
So, this is the required answer.

Note- In such types of questions first calculate the number of ways of sitting boys and girls respectively such that girls occupy only corner seats then multiply the number of ways of sitting boys and girls, we will get the required number of ways of sitting such that girls occupy only corner seats.