I like the theorem, I think it's Gallagher's, that says: Most polynomials with integer coefficients are irreducible and have the full symmetric group as Galois group (over the rational numbers).
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).
