The precise formulation asserts that the number of bad polynomials, i.e., the number of polynomials $X^r + a_1 X^{r-1} + \cdots + a_r$ with $|a_i|\leq N$ that DO NOT have the full symmetric group as Galois group is $$O(r^3(2N+1)^{r-\frac{1}{2}}\log N)$$ (out of $(2N+1)^r$ polynomials).