In this previous question, it was asked how many different ways we can arrange 4 managers and 3 employees in 7 seats around a circular table. One user said that there were 144 ways. I said there were 16. Whose answer is correct? Are we both wrong? Is it dependent on what is meant by "arrange"?

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    $\begingroup$ Not the way we do things here. $\endgroup$ – Gerry Myerson Sep 17 '13 at 12:54
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    $\begingroup$ I don't get it: what's the OP supposed to do? To write a comment in his own old question, a comment very probably not many (or very few, in fact) people will see as it is a past question, and remain with the doubt? I think this is valid way to post a valid mathematics question, and unless someone explains to me why this is wrong I'm voting to reopen. $\endgroup$ – DonAntonio Sep 17 '13 at 13:08
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    $\begingroup$ No reason to serial downvote. There are times when answers to a question conflict, and voting doesn't indicate any concensus on "which if any is right". So passerby-ers are likely to be confused, if they have the same question. So what to do? repost and risk being slammed with "DUPLICATE", or be straight and upfront with a link to the post in question and a point blank question regarding: "so what is correct here." Granted, I'd have rewritten the question itself with the conflicting answers and a link to the post in question, but cut some slack folks. $\endgroup$ – Namaste Sep 17 '13 at 13:09
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    $\begingroup$ @DonAntonio Downvotes aren't such a bad thing, they are no death sentence, or even a sentence to anything. I wasn't angry with Atul, nor, I believe, was anyone else. I just firmly believe that questions should be immanent since content on other questions might change – what if the other guy or girl deleted his or her answer? So therefore this question was a bad question for it didn't contain a real question. Now, it does contain a real question. Therefore I undownvoted. $\endgroup$ – k.stm Sep 17 '13 at 13:49
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    $\begingroup$ A simple edit to the original question would bring it back to the front page. An edit pointing to the disagreement would have accomplished the same purpose as posting this non-question. $\endgroup$ – Gerry Myerson Sep 17 '13 at 13:54

It depends on what you mean by arrange. Do you think all managers are interchangeable and all employees are interchangeable, or is an arrangement where you swap the seats of two employees different? Are the seats numbered, so if we rotate everybody one place we get a different arrangement? How about if we mirror the arrangement, so clockwise becomes counterclockwise-is that different?

If the people are interchangeable and the seats are not numbered, we must have a subsequence $EMME$ because two managers must sit together (there aren't enough employees to separate them) and a third is not permitted. The only arrangements are then $MMEMMEE, MMEMEME$-$2$ of them. If the seats are numbered, each of these can be rotated seven ways, giving $14$ arrangements. If the seats are not numbered but the people are distinguishable, we can arrange the managers $4!=24$ ways and the employees $3!=6$ ways for each order, giving $2\cdot 24 \cdot 6=288$ ways. Finally if the people are different and the seats are numbered, we multiply by $7$ to get $2016$ arrangements.


For the record, the correct answer is $288$ (nobody had found it before I just answered the original question; Ross Millikan was close but his arithmetic fell prey to the gravitational attraction of a previous wrong answer).

  • $\begingroup$ Actually, the answer is 144 or 288 depends on whether you consider mirrored arrangement is same arrangement or not. $\endgroup$ – achille hui Sep 22 '13 at 15:14
  • $\begingroup$ @achillehui: That is true, but I don't think anybody interpreted the question in that way. Also note that if counting just subsets of $4$ of the $7$ places modulo symmetry, including reflection symmetry does not divide the number by$~2$. $\endgroup$ – Marc van Leeuwen Sep 22 '13 at 17:17

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