# In how many ways can 10 married couples line up for a photograph if every wife stands next to her husband?

In how many ways can $10$ married couples line up for a photograph if every wife stands next to her husband?

I've given this a shot, now I just wanna compare my answers to see if I'm correct.

My answer is $2 \times 10!$ , the $2$ is because you can have either the husband standing first then the wife and vise versa. the $10!$ is the ways the arrange each couple.

• The basic idea is ok. But you need a factor 2 for each couple. – almagest May 3 '16 at 18:12
• But there are $10$ couples, not $5$, and for each of them you have to decide on the order. – lulu May 3 '16 at 18:12

First, label the couples $1$ through $10$ and choose an order for them in $10!$ ways. Then, for each of the ten couples, choose whether the wife or the husband goes before in $2$ ways, for a total of $2^{10}\cdot10!$ ways.
You're close, but not quite correct. First of all there are 10 couples and not only 10 persons. Then note that each couple can choose whether wife or husband is first independently. Therefore you get $10!2^{10}$ possible arrangements.