For the binding of peptides to wild-type HIV-1 and BIV TAR RNA and to mutants with bulges of various sizes, changes in the ΔΔG values of binding were determined from experimental Kd values. The corresponding entropies of these bulges are estimated by enumerating all possible RNA bulge conformations on a lattice and then applying the Boltzmann relationship. Independent calculations of entropies from fluctuations are also carried out using the Gaussian network model (GNM) recently introduced for analyzing folded structures. Strong correlations are seen between the changes in free energy determined for binding and the two different unbound entropy calculations. The fact that the calculated entropy increase with larger bulge size is correlated with the enhanced experimental binding free energy is unusual. This system exhibits a dependence on the entropy of the unbound form that is opposite to usual binding models. Instead of a large initial entropy being unfavorable since it would be reduced upon binding, here the larger entropies actually favor binding. Several interpretations are possible: (i) the higher conformational freedom implies a higher competence for binding with a minimal strain, by suitable selection amongst the set of already accessible conformations; (ii) larger bulge entropies enhance the probability of the specific favorable conformation of the bound state; (iii) the increased freedom of the larger bulges contributes more to the bound state than to the unbound state; (iv) indirectly the large entropy of the bound state might have an unfavorable effect on the solvent structure. Nonetheless, this unusual effect is interesting.