Step 1: Molecule X. The two hydrogens \(H_a\) and \(H_b\) lie on a carbon in a highly symmetric (meso/C\(_2\)) environment bearing identical \(\ce{HO-}\) and \(\ce{-CO2H}\) substituent sets on opposite sides. A \(C_2\) rotation superposes \(H_a\) and \(H_b\); replacing either gives the \emph{same} molecule \(\Rightarrow\) homotopic.
Step 2: Molecule Y. In the bridged/aromatic framework, \(H_a\) and \(H_b\) are related by an improper/mirror operation: replacement of \(H_a\) versus \(H_b\) generates non-superposable mirror-image products, while the parent is achiral. Thus they are enantiotopic.
Step 3: Molecule Z. For the prochiral vinyl \({CH_2}\) adjacent to \(\ce{Cl}\), the two vinylic hydrogens are in an enantiotopic relationship (replacing one or the other creates enantiomeric \(E/R_e\) vs \(S_i\) labeled products). Hence enantiotopic.
Step 4: Collecting: \(X\) homotopic, \(Y\) enantiotopic, \(Z\) enantiotopic \(\Rightarrow\) option (B).