We say that permutations π1, . . ., πr ∈ Sn invariably generate Sn if, no matter how one chooses conjugates π'1, . . ., π'r of these permutations, the π'1, . . . m π'r permutations generate Sn. We show that if π1,π2, and π3 are chosen randomly from Sn, then, with probability tending to 1 as n→∞, they do not invariably generate Sn. By contrast, it was shown recently by Pemantle, Peres, and Rivin that four random elements do invariably generate Sn with probability bounded away from zero. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.
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