Question

In how many orders can four girls and four boys walk through a doorway single file if

(a) there are no restrictions?
(b) the girls walk through before the boys?

Answer 1

(a) 40,320

(b) 576

Explanation:

(a) We are asked to find the number of orders in which four girls and four boys can walk through a doorway single file if there are no restrictions.

Since there are 8 people in total, so they can walk in 8! ways.

`8!=8\cdot7\cdot6\cdot5\cdot4\cdot 3 \cdot 2\cdot1=40,320`

Therefore, four girls and four boys walk through a doorway single file in 40,320 ways.

(b) To find the number of ways in which four girls and four boys can walk through a doorway single file if the girls walk through before the boys.

The total number of ways would be equal to product of 4 girls walking first and 4 boys walking after.

4 girls can walk first in 4! ways and boys can walk after in 4! ways as well.

`4!\cdot 4!=4\cdot 3\cdot 2\cdot 1\cdot 4\cdot 3\cdot 2\cdot 1=576`

Therefore, four girls and four boys can walk through a doorway single file if the girls walk through before the boys, in 576 ways.