# Thread: 8 people around a table

1. ## 8 people around a table

I know that they can be arranged a possible 5040 ways (7!), but could somebody help me, I'm not sure how to work out how they can be rearranged again, but with nobody sitting in the same seat as before? Help would be great! Thanks

2. ## Re: 8 people around a table

Maybe this is relevant.

3. ## Re: 8 people around a table

Originally Posted by Davidj
I know that they can be arranged a possible 5040 ways (7!), but could somebody help me, I'm not sure how to work out how they can be rearranged again, but with nobody sitting in the same seat as before? Help would be great!
I agree with reply #2: $\displaystyle D(8)=14833$.
Now it may seem absurd to you that the answer is more than the answer you gave. But there is a reason.

Suppose that each chair has one and only one of the eight people's name on it, as on a movie set. Now there are $\displaystyle 7!=5040$ different ways to arrange those chairs in a circle. BUT there are $\displaystyle 8!=40320$ different ways for the eight people to take one of the eight seats. There are $\displaystyle D(8)=14833$ ways for no one to take a seat with her/his name on the back of the seat.