We show that the Galois group of the polynomial in the title is isomorphic to the full symmetric group on six symbols for all but finitely many $n$. This complements earlier work of Filaseta and Moy, who studied Galois groups of $\binom{ n}{ 0} + \binom{ n}{ 1} X + \ldots + \binom{ n}{ k} X^k$ for more general pairs $(n,k)$, but had to admit a possibly infinite exceptional set specifically for $k=6$ of at most logarithmic growth in $n$. The proof rests upon invoking Faltings' theorem on a suitable fibration of Galois resolvents to show that this exceptional set is, in fact, finite.
Categories: math.NT