  Sharp, nonasymptotic bounds are obtained for the relative entropy between the
distributions of sampling with and without replacement from an urn with balls
of $c\geq 2$ colors. Our bounds are asymptotically tight in certain regimes
and, unlike previous results, they depend on the number of balls of each colour
in the urn. The connection of these results with finite de Finetti-style
theorems is explored, and it is observed that a sampling bound due to Stam
(1978) combined with the convexity of relative entropy yield a new finite de
Finetti bound in relative entropy, which achieves the optimal asymptotic
convergence rate.
