Discrete Math: Seating at a circular table - Possible Problem and My thinking:
Imagine a circular table, and you want to sit 7 people around it. The total arrangements would be 7!/7 or 6!. So, the order of left or right does not matter because we can sit these people anywhere and in any direction we want.
However, If they are denoted A, B, C, D, E, F, G. Then the order matters. Thus it would be 2*(7!/7) or 2*(6!).
- Logical Questions:
Am I right in my logic? How should I know the order of the left and right matters?

Also I realized something:
If two people, assuming A and B individuals insist on sitting next to each other in a circular table of 7 people. Then now we should keep A and B together while we still can switch the places of C D E F G people. In another words the arrangements would be 2*(5!) because A to the right of B (BA) and B to the right of A (AB). The arrangements can go clockwise and counter clockwise. 
