I have the following exercise: "Find number of ways to sit n people in a long table with n seats":
a. Everyone is seated on one-side of table: This case it's
b. We assume seating left to right is the same as seating right to left (seating A, B, C, ... from left to right is the same as from right to left, we only care about who each person's neighbor is): This case it's
n!/2 since we exclude the overlap cases in part a.
c. n is even and half of the people are seated on each side of table: This case I believe it is
2 * (n/2)! since there are n/2 seats and n/2 people in each side.
d. We seat people on both sides as in (c) and all we care about is who a person’s neighbors are on each side, as in (b).
e. We are dealing with a seating as in (d), but now we also care about who is sitting opposite a person as well as who a persons neighbors on each side are.
I would really appreciate if anyone could verify if my answers for part a, b, c are correct as well as give me hints for part d and e.