DevB 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 total number of different possible seating arrangements for the group?
1. 5
2. 10
3. 24
4. 32
5. 120
For circular permutations, we generally count arrangements RELATIVE TO THE FIRST PERSON SEATED.
Step 1: Place person A at the table.
Step 2: Count the number of ways to arrange the REMAINING people.
In the problem above, after person A is seated, the number of ways to arrange the remaining 4 people = 4! = 24.
The correct answer is
C.
Another problem:
At a dinner party, Amy, Bob, Cindy, David, Ellen, and Frank 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 total number of different possible seating arrangements for the group if Amy must sit directly across from Bob?
After Amy is seated:
Number of options for Bob = 1. (He must sit directly across from Amy.)
Number of ways to arrange the remaining 4 people = 4! = 24.
To combine these options, we multiply:
1*24 = 24.
Another:
At a dinner party, 3 men and 3 women 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 total number of different possible seating arrangements for the group if no man may sit adjacent to another man?
After man A is seated:
Number of women who could be seated to the right of man A = 3.
Number of remaining women who could be seated to the left of man A = 2.
Number of remaining women who could be seated directly across from man A = 1.
Number of ways to arrange the remaining 2 men = 2! = 2.
To combine these options, we multiply:
3*2*1*2 = 12.
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