Problem Solving

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 the people are different relative to each other. What is the number of different seating arrangements for the group?

A) 5
B) 10
C) 24
D) 32
E) 120

gdshamain wrote: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 the people are different relative to each other. What is the number of different seating arrangements for the group?

A) 5
B) 10
C) 24
D) 32
E) 120

In a CIRCULAR arrangement, the first person seated is irrelevant.
Our job is to count the number of ways to arrange the REMAINING PEOPLE relative to the first person seated.
In the problem above:
Once 1 of the 5 people is seated, the number of ways to arrange the remaining 4 people = 4! = 24.

The correct answer is C.

For those who like formulas:
The number of ways to arrange n elements in a circle = (n-1)!.

For those wondering why we do this when we're arranging people in a circle but not when we're arranging them along a line, consider these three circles:



Seem different, right? But they aren't! A always has B on his right, D on his left, and C across from him, so these are simply three rotations of the same arrangement.

To correct for this, we essentially view all the arrangements from one person's perspective: we park him in one seat, then arrange everyone else around him. Arranged relative to A all the arrangements will be unique, so we subtract one person from the total before factorializing.