I think I've solved it . . .
Two couples, Mr & Mrs A and Mr & Mrs B and 8 other people are to be randomly seated
at a round table. In how many of the possible 11! circular arrangements will Mr and Mrs A
sit directly opposite each other and Mr and Mrs B do not sit next to each other.
Mr. A can take any seat . . . it doesn't matter.
Mrs. A must take the opposite seat (one choice).
The other 10 people can be seated in ways.
. . Hence, there are 10! ways that the A's are directly opposite each other.
Of these 10! arrangements, how many have the B's together?
The A's are already seated opposite each other.
Duct-tape the B's together.
Then we have 9 "people" to seat.
There are 9! ways for them to be seated.
. . Hence, there are 9! ways for the B's to be together (and the A's are opposite).
Therefore, there are: . ways
. . that the A's are opposite and the B's are not together.