Assuming rotations do not matter (i.e. if we number the seats {1,2,3,4,5} then
{A,B,C,D,E} is the same as {B,C,D,E,A} ) then the number of ways is:
(5-1)! = 4! = 
[ If the seat number that a person is sitting in does matter (say there is a special head-of-the-table seat), then there are 5! = 120 ways for them to be seated, basically the result above but multiplied by 5 for the 5 head-of-table seatings ]
