so i'll assume that you got 120 by multiplying 5 x 4 x 3 x 2 x 1. this would be the correct answer if the 5 people were seated in a row, so that every possible ordering would be different from every other possible ordering.

however, the people are seated around a circular table, so that certain orderings can be rotated to produce other, ostensibly "different" orderings.
for instance, let's arbitrarily call one chair the "head" of the table. if the ordering, starting at the head of the table and going, say, clockwise, is ABCDE, that's indistinguishable from BCDEA (or CDEAB, or DEABC, or EABCD).

so, there are going to be fewer than 120 orderings that are actually distinct.

--

2 ways of solving the problem:

(1) divide by a redundancy factor:
realize that, because of the aforementioned rotation, every arrangement will be repeated 5 times if you count 5x4x3x2x1 = 120 possibilities.
because every distinct arrangement is actually counted five times, divide by five to find the number of truly distinct arrangements: 120 / 5 = 24.

(2) fix one chair:
you can ensure that each ordering is genuinely different by fixing the seating place of one of the five people. (this way, you know that your solutions can't be rotated into any of the other possible solutions, because the "fixed" person would wind up in the wrong place.)
so, let's say "A" must be listed first.
in this case, then, you're only choosing the seats for the remaining 4 people (B, C, D, E). this can be done freely, so that the number of arrangements is 4 x 3 x 2 x 1 = 24.