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In how many ways can 7 people be seated at a round table if they can sit anywhere?

The answer is 6!=720

I expected 7! I don't understand why. Does it have anything to do with the fact that the table is round?

TSR
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    Yes. Seat one person. How many ways can you seat the people on the right hand side? – For the love of maths May 02 '19 at 17:10
  • See https://math.stackexchange.com/questions/289532/arrangement-around-a-circular-table – PurpleCube May 02 '19 at 17:12
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    As stated the question is a little ambiguous, which is probably causing your confusion. The implication of stating that the table is round is that rotations are considered to be the same seating arrangement. It would be better if that implication were made express. – Robert Shore May 02 '19 at 17:12

2 Answers2

6

Imagine 7 people sat around a table, then ask everyone to move to the seat to their right, and to do this again and again and again - all of these arrangements would be the same, it's essentially just the circle which has been rotated, there is no difference in the arrangement! These people could keep doing this, moving to the seat to their right until they're back to the 'seat' they started at, this would happen 7 times, so we divide by this 7 as these are not unique combinations.

In general, for n-seats there are (n-1)! ways

Dinglevak
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If there are n people, there are (n-1)! ways to them to sit around a table.

So, there are (7-1)!=720;ways

Nadun Rajapaksha
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