Hello, Chris!

IthinkI'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'stogether?

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 arenottogether.