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Bunuel
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Bunuel
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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!

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
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find-the-number-of-ways-in-which-four-men-two-women-and-a-106919.html

Hope it helps.
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Bunuel
If 5 noble knights are to be seated at a round table, then how many different ways can they be seated?

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


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

VisualRepresentation_SeatingAroundTable.png
VisualRepresentation_SeatingAroundTable.png [ 26.85 KiB | Viewed 16832 times ]

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

P.S. To elaborate on what the last line, "two seating arrangements are considered different only when the positions of the people are different relative to each other,’ means:

It means that rotating the same arrangement around the table doesn’t count as a new one. Only when the knights’ positions change relative to one another is it considered a different arrangement.

For example, if A, B, C, D, and E are seated in that order around the table, rotating them to B, C, D, E, A is the same arrangement because everyone has the same neighbors. But if their relative order changes, like A, C, B, D, E, that’s a different arrangement.
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