Answer:
(a) 40,320
(b) 576
Step-by-step 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.
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.
Therefore, four girls and four boys can walk through a doorway single file if the girls walk through before the boys, in 576 ways.