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Sara has $6$ flower pots, each having a unique flower. Pots are arranged in an arbitrary sequence in a row. Sara rearranges the sequence each day but no two pots should be arranged adjacent to each other, if they have already been adjacent to each other. For how many days can Sara keep rearranging the flowers (i.e. how many arrangements satisfying the rule above are possible)?

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  • $\begingroup$ How did you approach the problem? $\endgroup$ – Gerard Nov 10 '13 at 4:11
  • $\begingroup$ I have a pretty naive approach, i mean we have like 6! possibilities. I am thinking of recursion but have no clue after that $\endgroup$ – user106448 Nov 10 '13 at 4:17
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It seems the following.

If I understood you right then we have ${6 \choose 2}=15$ different pairs of pots. At each days Sara realizes 5 of these pairs as ajacent. Since all these pairs should be different, the number of days is at most $15/5=3$ . The following example describes the admissible list of arrangements for 3 days.

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