Le us name the couple aa,bb,cc,dd. The husband of coupl aa is a, the wife of aa is a. The members of the other couples are named similar.
The places of the desk are numbered (clockwise) 1,2,3,...,8 (and next to 8 comes 1 again).
You have $8$ possibilities to reservate places for the first couple.
- e.g. we place couple aa clockwise starting with seat 4:
- the first couple is placed on the seats 4,5
You have $3!$ possibilities to specify the order of the couples following couple one in a clockwise sense.
- after aa comes cc then bb then dd:
- so cc has assigned 6,7
- bb has assigned 8,1
- dd has assigned 2,3
Then four each of the 4 couples you have 2 possibility to placethe husband and the wife on the seats assigned to the couple. So these are $2^4$ possibilities.
All in all you have
$$8\cdot 3! \cdot 2^4=768$$
Our special selection is
1 2 3 4 5 6 7 8
b D d A a c C B
If one does not distinguish between arrangement that differ only by one of the 8 rotations then there are only