M01-24

Question

Five noble knights are to be seated at a round table. How many different seating arrangements are possible, given that two seating arrangements are considered different only when the positions of the people are different relative to each other?

A. 120
B. 96
C. 60
D. 35
E. 24

A. 120
B. 96
C. 60
D. 35
E. 24

Bunuel · Accepted Answer

Official Solution:

Five noble knights are to be seated at a round table. How many different seating arrangements are possible, given that two seating arrangements are considered different only when the positions of the people are different relative to each other?

A. 120
B. 96
C. 60
D. 35
E. 24

A. 120
B. 96
C. 60
D. 35
E. 24

• The number of arrangements of  n  distinct objects in a row is given by  n! .
 
• The number of arrangements of  n  distinct objects in a circle is given by  (n-1)! .
 
Why this difference? In a row, if you shift all objects by one position, you get a new arrangement. But in a circle, this shift results in the same relative arrangement. That's why the number of distinct arrangements for a circle is:
 
•  \frac{n!}{n} = (n-1)! .
 
For 5 knights, that is:
 
 (5-1)! = 4! = 24 .
 
Therefore, there are 24 different seating arrangements for the knights.

Answer: E

OutOfTheHills · Answer

Hi Bunuel,

I'm having trouble visualizing and grasping this question. I got my answer by calculating 5! instead of 4!. Would you mind explaining why arranging order in a line (which we would use 5!) is different from arranging order in a circle? Thanks!

Bunuel · Answer

The number of arrangements of n distinct objects in a row is given by n! . The number of arrangements of n distinct objects in a circle is given by (n-1)! . The difference between placement in a row and that in a circle is following: if we shift all object by one position, we will get different arrangement in a row but the same relative arrangement in a circle. So, for the number of circular arrangements of n objects we have: \frac{n!}{n} = (n-1)! . Theory on Combinations: math-combinatorics-87345.html DS questions on Combinations: search.php?search_id=tag&tag_id=31 PS questions on Combinations: search.php?search_id=tag&tag_id=52 Tough and tricky questions on Combinations: hardest-area-questions-probability-and-combinations-101361.html Similar questions to practice: seven-men-and-seven-women-have-to-sit-around-a-circular-92402.html a-group-of-four-women-and-three-men-have-tickets-for-seven-a-88604.html the-number-of-ways-in-which-5-men-and-6-women-can-be-seated-94915.html in-how-many-different-ways-can-4-ladies-and-4-gentlemen-be-102187.html 4-couples-are-seating-at-a-round-tables-how-many-ways-can-131048.html at-a-party-5-people-are-to-be-seated-around-a-circular-104101.html seven-family-members-are-seated-around-their-circular-dinner-102184.html seven-men-and-seven-women-have-to-sit-around-a-circular-11473.html a-group-of-8-friends-sit-together-in-a-circle-alice-betty-106928.html seven-men-and-five-women-have-to-sit-around-a-circular-table-98185.html a-group-of-8-friends-sit-together-in-a-circle-alice-betty-106928.html find-the-number-of-ways-in-which-four-men-two-women-and-a-106919.html Hope it helps.

TheNightKing · Answer

One Approach : Remember this for a Circular arrangement : n people can be seated in (n-1)! ways in case of a round table or round anything. So in this case that would be 4!=24. Option E. Second Approach : Let's say you don't know the first approach. Now think of you walking into a round table at a restaurant with 5 chairs. How many ways you can sit? Think really good about it. There is only one way and not five ways because only and only after you sit the other chairs can be positioned with according to your position. So the first person has only one way to sit on the table and second will have now 4 ways the third will have 3 ways and so on. 1*4*3*2*1=24 ways. P.S: You cannot argue that in the restaurant depending on which seat you chose you will get a different view. That's not the argument here. The argument is just the table and you cannot assume your own little world

AnirudhaS · Answer

I have added a helpful visual representation for people who are not getting the hang of the concept.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

kouassidrissa047 · Answer