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I want to understand how to arrange $12$ people around a circular table with $7$ chairs. We don't care about the overflow, those people can go to another table.

I thought the way to solve the problem is that the position for the first chair is fixed, the second chair has $11$ possible options of people (since one person is already seated), the third chair has $10$ possible options, the fourth chair has $9$ possible options and so on until we get to the seventh chair which has $6$ possible options of people.

So I thought the way to solve is that this is a permutation problem $1*11*10*9*8*7*6=332640=11P6=\frac{12P7}{12}$. But my professor says the correct answer is $\frac{12P7}{7}$. I don't understand why we should divide $12P7$ by the number of chairs. Can someone explain this me?

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    $\begingroup$ Choose which seven people get a seat. Then, let the youngest of those people sit down first at the table wherever they like. Then, fill the remaining six of the seven chosen people around the table. $\endgroup$ – JMoravitz Mar 10 at 20:07
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I think it should be $$\binom{12}{7}6!$$

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  • $\begingroup$ Why multiply by $6!$? $\endgroup$ – Sam Mar 10 at 20:14
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    $\begingroup$ @Sam Because we take one of these seven and for the rest we make all permutations. We think, of course, that any cyclic permutations of these seven are the same. $\endgroup$ – Michael Rozenberg Mar 10 at 20:16
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    $\begingroup$ @Sam Note that $\binom{12}{7}6!$ is equal to $\frac{_{12}P_7}{7}$. I much prefer this answer (which matches mine above in the comments) as it avoids the "division by symmetry" style arguments that seem common and confuses people. $\endgroup$ – JMoravitz Mar 10 at 20:30
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If the only important thing while arranging places is the locations of people relative to each other; or in other word, if nothing changes when we rotate all the chairs together around the circular table, according to the problem, professor is right. However, if the positions of chairs are important for us, such as sitting around circular table in a conference (Somebody must be turns behind to the scene), it would not be different from line-permutation .

Dividing by the number of chairs is comes from the rotation. If you rotate all chairs together 1,2,3,...,n times, it does not change location of people relative to each other, but it does relative to room, scene, board etc.

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