Question

Barbara and John and six other people go through a doorway one at a time. Find the number of ways in which the eight people can go through the doorway if John goes through the doorway after Barbara?

I know the answer is `(8!)/2 = 20160`, but I'm not sure how to get to it.

Can anyone please just quickly explain how to get `(8!)/2`?? No lengthy explanation needed.

Thank you!

Answer 1

See below.

Explanation:

With 8 people walking through the day one way at a time, since order matters, there are 8! ways for the people to be arranged through the door.

Since only one person can go through the door at a time, either John goes through after Barbara or Barbara goes through after John. They can't go through together.

Because of that, exactly half of the possible combinations will be the kind we want (John after Barbara)) and the other half don't fit our problem. Thus, we are asking about half of all possible arrangements.

`(8!)/2 = 20160`