At a dinner party, 5 people are to be seated around a circular table. Two seating arrangements are considered different only when the positions of people are different relative to each other. What is the total no. of different possible seating arrangements for the group?
Ans: 24

I think I need to really polish my propbability skills... I really need help on this one. I guess, I am not getting the concept right. Can you please suggest me how this should be solved?

Cheers
~a

fix one and rest can be arranged in 4! ways that is 24

well put.

the difference between a round table and a normal table is that a round table has no ends. therefore, you can arbitrarily select which seat is to be called seat #1.
two ways to approach this problem:

(1)
as the poster above has said, you can fix one of the people in place (or, equivalently, rotate the table like a carousel so that that person always winds up in the same place). this will avoid the production of multiple equivalent scenarios in which the people are seated in the same order, but have just shuffled a seat or two over (these don't count as different arrangements).
then you have free rein to arrange the other four people, so that's 4! = 24 arrangements.

(2)
alternatively, you can figure out that there are 5! = 120 arrangements overall.
however, each UNIQUE arrangement is actually repeated five times: because there are five seats at the table, there are 5 different versions of every possible seating arrangement. (for instance, ABCDE, BCDEA, CDEAB, DEABC, EABCD are all equivalent.)
so this means you must divide by 5: 120 / 5 = 24

I have been reviewing this problem, and I think maybe I don't understand understand the meaning of "different relative to each other." Can you explain to me why each unique arrangement is repeated five times, and not ten? For the example Ron gave:

ABCDE, BCDEA, CDEAB, DEABC, EABCD but not
EDCBA, AEDCB, BAEDC, CBAED, DCBAE

To me, placing people in the same order, whether clockwise or counterclockwise, seems to be the same position relative to each other. Everyone is sitting next to the same two people.

My answer to this question, on the test and in review, is perpetually "12." What am I missing here?

if you are to my left and my wife is to my right (first five arrangements), that's a different arrangement than if you are to my right and she is to my left (last five).