  We prove a result on the asymptotic proportion of randomly chosen pairs of
permutations in the symmetric group $S_n$ which "invariably" generate a
nonsolvable subgroup, i.e., whose cycle structures cannot possibly both occur
in the same solvable subgroup of $S_n$. As an application, we obtain that for a
large degree "random" integer polynomial $f$, reduction modulo two different
primes can be expected to suffice to prove the nonsolvability of
$Gal(f/\mathbb{Q})$.
