Now, imagine a specific arrangement. Now, have everyone get up and move one seat to their right. That arrangement was counted separately from the first one in the 8! but it’s really the same since if each person looks to their left and their right, they’re sitting next to the same person. There are 8 such “identical rotations” for each possible seating arrangement so the total number of seatings is 8! / 8 = 7! = 5040.
