Which of the following salts has the greatest molar solubility in pure water?

For CaCO₃, solubiliry can be calculated as
$$\begin{gathered} {\mathrm{CaCO}}_3\rightleftharpoons {\mathrm{Ca}}_{\mathrm{S}}^{2+}+\underset{\mathrm{S}}{{\mathrm{CO}}_3^2} \\ {\mathrm{K}}_{\mathrm{sp}}=\mathrm{S}\cdot \mathrm{S}={\mathrm{S}}^2\Rightarrow \mathrm{S}=\sqrt{{\mathrm{K}}_{\mathrm{sp}}} \\ =\sqrt{8.7\times {10}^{-9}}=0.0051=9.3\times {10}^{-5} \end{gathered}$$
For CuS, solubility can be calculated as
$$\begin{gathered} \mathrm{CuS}\rightleftharpoons {\mathrm{Cu}}^{2+}+{\mathrm{S}}_{\mathrm{S}}^{2-} \\ {\mathrm{K}}_{\mathrm{sp}}=\mathrm{S}\cdot \mathrm{S}={\mathrm{S}}^2 \\ \mathrm{S}=\sqrt{{\mathrm{K}}_{\mathrm{sp}}}=\sqrt{8.5\times {10}^{-45}}=\sqrt{85\times {10}^{-46}} \\ =\sqrt{85}\times {10}^{-23}=9.21\times {10}^{-23} \\ \text{ For }{\mathrm{Ag}}_2{\mathrm{CO}}_3,\text{ solubility can be calculated as} \\ {\mathrm{Ag}}_2{\mathrm{CO}}_3\rightleftharpoons {\underset{2\mathrm{S}}{2\mathrm{Ag}}}^{+}+\underset{\mathrm{S}}{{\mathrm{CO}}_3^{2-}} \\ {\mathrm{K}}_{\mathrm{sp}}=(2\mathrm{S}{)}^2\cdot \mathrm{S}=4{\mathrm{S}}^3\Rightarrow \mathrm{S}=\sqrt[3]{\frac{{\mathrm{K}}_{\mathrm{sp}}}{4}} \\ \mathrm{S}=\sqrt[3]{\frac{6.2\times {10}^{-12}}{4}} \\ =\sqrt[3]{1.55\times {10}^{-12}}=1.15\times {10}^{-4} \end{gathered}$$
For existence of Pb₉IO₃, there must be ${\mathrm{IO}}_3^{-}\text{and }{\mathrm{Pb}}_9^{+}$ present as constituent ions but in real ${\mathrm{Pb}}_9^4-$ exist as ion. When we match the K_sp value of iodate of lead, we found that compound must be $\mathrm{Pb}{\left({\mathrm{IO}}_3\right)}_2.$
Now, we can calculate solubiliry of compound as
$$\begin{gathered} \mathrm{Pb}{\left({\mathrm{IO}}_3\right)}_2\rightleftharpoons {\mathrm{Pb}}_{\mathrm{S}}^{+}+2{\mathrm{IO}}_3^{-} \\ {\mathrm{K}}_{\mathrm{sp}}=\mathrm{S}\cdot (2\mathrm{S}{)}^2=4{\mathrm{S}}^3 \end{gathered}$$
$$\begin{gathered} \mathrm{S}=\sqrt[3]{\frac{{\mathrm{K}}_{\mathrm{sp}}}{4}}=\sqrt[3]{\frac{2.6}{4}\times {10}^{-13}} \\ =\sqrt[3]{\frac{260}{4}\times {10}^{-15}} \\ =\sqrt[3]{65\times {10}^{-5}}=4.02\times {10}^{-5} \end{gathered}$$
Hence, maximum solubility has been observed for (c), i.e.
${\mathrm{Ag}}_2{\mathrm{CO}}_3$