Richard Lee wrote:
At a dinner party , 5 people are to be seated around a circular table . Two seating arrangments are considered different only when the positions of the people are different relative to each other. What is the total number of different possible seating arrangments for the group?
24. We have to fix one person so its actually 4! I'm not really sure why we have to fix one person. I just know we do. Some more insight on the reasoning behind that would be awsome.
Consider 5 points chairs on a straight line (A..E) each with one person seated on it. Push each person to the seat next to him. The last one (E) will move to seat A which changes the arrangement (as the relative position of E changes).
Now consider the same 5 chairs and people on a circle. In this case E already has A as his neighbor. moving a person to the chair next to him does not change the arrangement. E still has A as his neighbor and there is no change in the relative position of A and E. Same is the case with rest of the people. A (and similarly B,C,D and E) can occupy one of the 5 chairs without changing their relative position and hence the arrangement for any particular seating sequence. Therefore we need to divide 5! by 5 to arrive at the final answer.
Hope this helps and apologies if it sounds convoluted.