Circular Permutation problem

In summary, there is a discussion about how many ways 10 people can sit around a roundtable if 3 particular people sit together. The solutions involve finding the number of ways to seat 8 people and then multiplying by 3! or finding the number of ways to pick three consecutive chairs on the roundtable and multiplying by 7!3!. It is mentioned that the interpretation of "roundtable" can affect the answer, as all seats are equal and the order matters.
  • #1
dumpman
17
0

Homework Statement


how many ways can 10 people sit around a roundtable if 3 particular people sit together



Homework Equations





The Attempt at a Solution


my attempt was (8-1)! x 3!
 
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  • #2
Hi dumpman! :wink:
dumpman said:
how many ways can 10 people sit around a roundtable if 3 particular people sit together

my attempt was (8-1)! x 3!

Looks ok to me! :smile:
 
  • #3
That would only be right if you designate 3 particular seats for the 3 particular people to sit in. I think you have a number of choices for those 3 particular seats.
 
  • #4
if 3 arent assigned specifically, then the answer is 10C3 x (8-1)! x 3! ?
 
  • #5
What you can do is think of those 3 as constituting one person. That is, find the number of different ways of seating 8 people rather than 10, then multiply by 3! for the number of ways to seat those 3 people. Oh, wait, that is (8-1)! 3!, your first answer.

Dick, that does NOT require designating "3 particular seats for the 3 particular people to sit in". Or are you suggesting we should multiply by 7!3! by 8 to allow for those 3 being anyone of the original "8" people? I don't believe that is correct.

(I could be wrong, now!)
 
  • #6
If the question means "order of seating" then I think you are right if 'right' and 'left' partners are distinguishable. If the question means "ways to seat" as in 'the chairs are numbered' then I think I'm right. It is a little ambiguous. I THINK you have to multiply 7!*3! by the number of ways to pick three consecutive chairs in a roundtable. And that's not 10C3, dumpman. They have to be adjacent. But if you think circular rotations are not important you can go with Halls and tiny-tims answer.
 
Last edited:
  • #7
all seats are equal

Dick said:
But if you think circular rotations are not important you can go with Halls and tiny-tims answer.

Yes, in my experience, "roundtable" questions always mean that the order (clockwise, say) matters, but not the actual seats …

if you're attending a dinner-party, all you're interested in is where you're sitting in relation to everyone else …

on an ordinary table, you could be stuck at the end, which is different, but on a roundtable all seats are equal. :smile:
 

1. What is a circular permutation problem?

A circular permutation problem is a type of combinatorial problem in which objects are arranged in a circle rather than a straight line. It involves finding the number of unique arrangements or combinations of objects in a circular order.

2. How is a circular permutation problem different from a linear permutation problem?

In a linear permutation problem, the order of objects in a straight line matters. However, in a circular permutation problem, the order of objects in a circle is considered the same as long as the relative positions of the objects are maintained. This means that rotations do not change the arrangement and are not counted as separate arrangements.

3. What is the formula for solving a circular permutation problem?

The formula for solving a circular permutation problem is n!/n where n is the number of objects. This is because in a circular permutation, n rotations of the same arrangement are considered the same. Therefore, we divide the total number of arrangements (n!) by the number of rotations (n) to get the unique arrangements.

4. Can you give an example of a circular permutation problem?

One example of a circular permutation problem is arranging 5 people around a circular table. The number of unique seating arrangements would be 5!/5 = 24, as rotating the same arrangement would not change the overall seating arrangement.

5. Are there any real-life applications of circular permutation problems?

Yes, circular permutation problems have various real-life applications, such as scheduling shifts for employees in a circular work setting, arranging seats in a circular theater, or organizing a round-robin tournament in sports. They are also commonly used in computer science for tasks such as generating unique passwords or shuffling elements in a circular array.

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