First Problem

There are n people in the room and they are sitting on a round table. All of them went out the room and came back. When they are sitting on the same round table again, they are sitting in a certain way such that the person sitting on their right side is not the same person as before. How many ways are there to sit in this way?

To me, it sounded like Menage problem at first but found out that this is not the same problem.

At first, I thought there would be $D_n$ number of ways to sit them in the way the problem asks for. But I thought that since they are sitting on a round table, any rotations of a sitting will be counted as one case. So, I changed it to $D_{n-1}$ ways.

Am I on the right track?

Second Problem

There are $n$ pair of pieces for a necklace. For each pair, they have the same color but different letters are written on the pieces. How many ways are there to make the necklace such that any adjacent two pieces do not have the same color?

For this one, it really confuses me, but sounds really similar to the non-sexist Menage problem. How should I tackle with it?

Thank you!

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I figured out the second problem. – user103192 Mar 13 '14 at 16:20
Please share your solution for later confused patrons of the site. – vonbrand Aug 7 '14 at 1:12