A question in my mathematics homework is as follows:

Find the number of ways in which seven men can sit at a round table so that all shall not have the same neighbors in any two arrangements.

My doubt is about the darkened part. Does it mean that everybody should have different neighbors in every arrangement or does it mean that not everybody should have same neighbors in every arrangement.

  • 1
    The sentence is ambiguous. It could be interpreted either way. – fixer1234 Jul 11 '17 at 6:36
  • a permutation is given by definition: here, 7 times 6 are the ways you can arrange them in tupies. You can create a table to view it. – Lambie Jul 31 '20 at 23:22

It means that you have to find the number of arrangements in which you can sit the seven men in such a way that in each arrangement each man has unique neighbours every time. If in the first arrangement, Jack has Mark and Luke as his neighbours then in every other arrangement that you make, Jack must not have Mark or Luke as neighbours again.

So yes it means that everybody should have different neighbours in every arrangement.

  • The sentence wording could also mean that some but not all could have the same neighbors. – fixer1234 Jul 11 '17 at 19:31

The literal meaning of it is that, given any two arrangements A and B, there should not be anyone who has the same neighbors in A that they have in B. But very likely, what is meant is that if, for two arrangements A and B, everyone has the same neighbors in A that they have in B, then those should not be counted as separate arrangements. Given that this is from a math class, it is reasonable to expect them to choose more careful wording.

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