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"?

• Not the way we do things here. – Gerry Myerson Sep 17 '13 at 12:54
• 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. – DonAntonio Sep 17 '13 at 13:08
• 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. – Namaste Sep 17 '13 at 13:09
• @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. – k.stm Sep 17 '13 at 13:49
• 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. – Gerry Myerson Sep 17 '13 at 13:54

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).
• @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$~2$. – Marc van Leeuwen Sep 22 '13 at 17:17