Question

Ten couples are waiting in line at a buffet table. How many different ways can they stand in line if each couple needs to stay together?

Answer 1

`10! xx 2 = 3,628,800 xx 2 = 7,257,600`

Explanation:

We can look at the ways we can arrange the 10 couples and then multiply by the number of ways each person within a couple can be arranged.

Couples

There are 10 couples. We can arrange them in `10!` ways (there are 10 couples that can be first in line, then 9 couples second in line, then 8 couples third in line, etc).

People within couples

There are 2 people that compose each couple. They can stand in one of 2 ways (person A on the left or person B on the left).

Bringing it together

We have `10! xx 2 = 3,628,800 xx 2 = 7,257,600` ways for them to stand together.

If that sounds incredibly big, keep in mind that if we had arranged all 20 people in any which way (and so ignoring keeping couples together), we'd have:

`20! ~= 2.43xx10^18` ways to arrange them.