  We consider the number of colors for the colorings of links by the symmetric
group $S_3$ of degree $3$. For knots, such a coloring corresponds to a Fox
3-coloring, and thus the number of colors must be 1 or 3. However, for links,
there are colorings by $S_3$ with 4 or 5 colors. In this paper, we show that if
a 2-bridge link admits a coloring by $S_3$ with 5 colors, then the link also
admits such a coloring with only 4 colors.
