CLN025, a peptide with only 10 residues, folds into a specific β-hairpin structure (this is referred to as "native structure"). Here we investigate the stabilization mechanism for CLN025 using our free-energy function F. F comprises two components, the hydration entropy and the component related to the energetic dehydration effect. The former component is calculated using the hybrid of the angle-dependent integral equation theory (ADIET) and our recently developed morphometric approach. The ADIET is a statistical-mechanical theory applied to a molecular model for water. The latter component is calculated in a simple but judicious manner accounting for physically the most important factors: the break of polypeptide-water hydrogen bonds and formation of polypeptide intramolecular hydrogen bonds upon structural change to a more compact one. We consider the native structure, compact nonnative structures newly generated, and a set of random coils mimicking the unfolded state. F and its components are calculated for all the structures considered. The loss of the polypeptide conformational entropy upon structural transition from the unfolded state to a compact structure is also estimated using a simple but physically reasonable manner. We find that the key factor is the water-entropy gain upon folding originating primarily from an increase in the total volume available to the translational displacement of water molecules in the system, which is followed by the reduction of water crowding. The amino-acid sequence of CLN025 enables it not only to closely pack the backbone and side chains including those with large aromatic groups but also to assure the intramolecular hydrogen bonding upon burial of a donor and an acceptor when the backbone forms the native structure. The assurance leads to essentially no enthalpy increase upon folding. The close packing brings a water-entropy gain which is large enough to surpass the conformational-entropy loss. By contrast, it is not possible for the design template of CLN025, GPM12, to realize the same type of structure formation. There are significantly many compact structures which are equally stable in terms of F, and due to the conformational-entropy effect, the unfolded state is favorably stabilized.