marcodonzelli wrote:
At a dinner party, 5 people are to be seated around a circular table. 2 seating arrangements 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 arrangements for the group?
A. 5
B. 10
C. 24
D. 32
E. 120
Although we can quickly apply the circular arrangement formula (i.e., number of ways to arrange n objects in a circle = (n - 1)!), we can also solve the question using the Fundamental Counting Principle (FPC, aka the slot method). In the process of doing so, you'll also learn WHY the circular arrangement formula worksFirst label the five chairs as follows:
We can seat the first guest in one of the
5 available chairs.
We can seat the next guest in one of the
4 remaining chairs.
We can seat the next guest in one of the
3 remaining chairs.
We can seat the next guest in one of the
2 remaining chairs.
We can seat the last guest in the
1 remaining chair.
So, the total number of ways to seat the guests = (
5)(
4)(
3)(
2)(
1) =
120 ways
The answer, however, is NOT E, because we have inadvertently counted every possible arrangement 5 times.
For example, the five arrangements shown here...
... are all the
same, because the relative positions of the five people are the same in each case.
Since we have counted each unique arrangement 5 times, we must divide
120 by 5 to get
24 possible arrangements
Answer: C
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