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For the symmetry group $S_n$ ($n\geq1$), how many permutations $\sigma$ exist with the property that $\sigma$ doesn't map any element of $\{1,\ldots,n\}$ to itself? I know I can try to do a counting argument for small $n$ (the first numbers of $n$ ($1,2,3,4$), calculated by hand, are $0,1,2,9$) but I would like to find a general method.

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marked as duplicate by Jyrki Lahtonen group-theory May 9 at 11:47

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    $\begingroup$ The relevant term here is Derangements $\endgroup$ – lulu May 9 at 11:23
  • $\begingroup$ Thanks! I haven't found this before posting this question! $\endgroup$ – RMWGNE96 May 9 at 11:24
  • $\begingroup$ No problem. At least you have tried to search before. Many people don't. $\endgroup$ – Dietrich Burde May 9 at 12:07