Seating people around a table (elementary counting technique) Eight people, including Abigail and Bethany, are to be seated at a square table so that two people are along each edge. Two seatings are considered distinct if, and only if, the ordering of people starting with Abigail and continuing clockwise around the table in one seating is distinct from that in the other seating.  How many distinct seatings are there so that Abigail and Bethany are not next to nor across from each other?
There are 8!/8 = 7! seatings of the eight people around the table. There are 2(7!/7) = 2(6!) seatings with Abigail and Bethany next to each other, and there are 2(7!/7) = 2(6!) seatings with Abigail and Bethany across from each other. So, there
\begin{equation*}
7! - 2(6!) - 2(6!) = 2160
\end{equation*}
seatings in which Abigail and Bethany are not next to nor across from each other.
I am using the Complement Principle.  Is this correct?
 A: The way I read the problem I think you are subtracting $2(6!)$ one too many times. Unrolling the argument I suppose it should look like the following.
Since you don't actually care which side of the table anyone is sitting at, (Since two seatings are the same if the ordering starting with Abigail and going clockwise is different) you can pretty much model your seating arrangements by thinking of the ordering only. Given that there are 7! choices for the ordering (since we can fix Abigail to be the first), which you got. Then you can seat Bethany next to her on the left or the right that's one $2(6!)$.
Now we need to deal with the situations of sitting across form each other. The issue here is that if you seat the people $A,\_,\_,B,\_,\_,\_,\_$ then this is a valid seating arrangement since you can place $A$ on the left corner of the table in which case $B$ is not sitting across from them. Same situation is when you get $A, \_ , \_ , \_ , B , \_ , \_ , \_ $ in which case you can seat $A$ on the right corner of the table and $B$ is not across. Thus if you only care about the order of the guests clockwise from $A$ then the only cases that need to be excluded are the ones when $A$ and $B$ are next to each other. The ones where they are across from each other can still be valid seating arrangements by rotating them.
Note: this does depend how one interprets sitting across.
I chose to interpret it as the situation where a line connecting to the two people is perpendicular to an axis of symettry of the table.
