Bhargava’s proof of van der Waerden conjecture: how to use Hilbert irreducibility to show almost all polynomials have Galois group $S_n$ On the first page of his paper where he proves van der Waerden’s conjecture, Bhargava mentions that Hilbert’s irreducibility theorem shows that the number of monic integer polynomials of degree $n$, all of whose coefficients have absolute value less than $H$, which have Galois group not equal to $S_n$ is $o(H^n)$. I suppose the argument he’s referring to is well-known or straightforward, although he does not cite a source. What is the argument?

The argument is old, going back to van der Waerden, with improvements by later authors such as Gallagher.  See here (look for the notation $E_n(N)$) and Theorem 1.2 here.
