Circular Permutation problem

AI Thread Summary
The discussion centers on calculating the number of ways 10 people can sit around a round table with the condition that 3 specific individuals must sit together. One proposed solution involves treating the 3 people as a single unit, leading to the formula (8-1)! x 3!. However, there is debate about whether specific seats need to be assigned to these individuals or if their arrangement can be flexible. Participants clarify that the arrangement should consider circular permutations, emphasizing that the order matters but the specific seats do not. The conversation concludes with the understanding that in roundtable scenarios, the relative positioning of individuals is more important than the exact seats.
dumpman
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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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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:
 
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.
 
if 3 arent assigned specifically, then the answer is 10C3 x (8-1)! x 3! ?
 
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!)
 
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:
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:
 

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