  In many situations, the monodromy group of enumerative problems will be the
full symmetric group. In this paper, we study a similar phenomenon on the
rational curves in $|\mathcal{O}(1)|$ on a generic K3 surface of fixed genus
over $\mathbb{C}$ as the K3 surface varies. We prove that when the K3 surface
has genus $g$, $1\leq g\leq 3$, the monodromy group is the full symmetric
group.
