Invariable generation of the symmetric group

We say that permutations π₁, . . ., π_r ∈ S_n invariably generate S_n 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 S_n, then, with probability tending to 1 as n→∞, they do not invariably generate S_n. By contrast, it was shown recently by Pemantle, Peres, and Rivin that four random elements do invariably generate S_n 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.