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I know the solution to this problem is (n-1)!. If it was a line we would have used n! for each spot as we care about order and we cannot choose a person more than once. I also know that the way you get to this number is because it is possible to rotate the circle through n possible combinations which are essentially the same. Here, i get a bit confused though. At this point, why not subtract these n (or maybe n times n for each person we choose to put at a specific chair) possibilities from the total? why do we divide by n?

p.s. I know about the explanation involving the first person as a pivot. I am mainly having a hard time with the division of n.

meroprog
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1 Answers1

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Try to take a smaller value for $n$, say $n=3$ and work through the possibilities.

For example, if the people are $a,b,c$, we would have $n! = 3! = 6$ possibilities: $$ a, b, c\\ a, c, b\\ b, a, c\\ b, c, a\\ c, a, b\\ c, b, a $$

If you consider them in a circle, the following rotated $n=3$ terms in each line will give rise to the same circular formation: $$(a, b, c), (b, c, a), (c, a, b)$$ $$(a, c, b), (c, b, a), (b, a, c)$$

Do you see where the division by $n$ comes in now?

user1952500
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