The correct match of the group reagents in List-I for precipitating the metal ion given in List-II from solutions, is

List-I	List-II
(P) Passing H₂S in the presence of NH₄OH	(1) Cu²⁺
(Q) (NH₄)₂CO₃ in the presence of NH₄OH	(2) Al³⁺
(R) NH₄OH in the presence of NH₄Cl	(3) Mn²⁺
(S) Passing H₂S in the presence of dilute HCl	(4) Ba²⁺
	(5) Mg²⁺

The correct option is P → 3; Q → 4; R → 2; S → 1.

To understand the correct matching, we analyze the systematic separation of basic radicals (cations) in qualitative inorganic analysis. Cations are classified into different groups based on the solubility products ($K_{\mathrm{sp}}$) of their salts (chlorides, sulfides, hydroxides, and carbonates) and are precipitated sequentially using specific group reagents.

1. Match for (P): Passing $H_{2}S$ in the presence of ${\mathrm{NH}}_4\mathrm{OH}$
This combination represents the group reagent for Group IV cations, which includes ${\mathrm{Mn}}^{2+}$, ${\mathrm{Zn}}^{2+}$, ${\mathrm{Ni}}^{2+}$, and ${\mathrm{Co}}^{2+}$.
In the presence of the weak base ${\mathrm{NH}}_4\mathrm{OH}$, the hydrogen ions from the dissociation of $H_{2}S$ are neutralized, shifting the equilibrium to produce a high concentration of sulfide ions ($S^{2-}$). This high concentration is sufficient to exceed the solubility products of Group IV metal sulfides, leading to the precipitation of manganese as manganese sulfide ($\mathrm{MnS}$).
Therefore, P matches with (3) ${\mathrm{Mn}}^{2+}$.

2. Match for (Q): ${\left({\mathrm{NH}}_4\right)}_2{\mathrm{CO}}_3$ in the presence of ${\mathrm{NH}}_4\mathrm{OH}$
This mixture is the group reagent for Group V cations, which includes ${\mathrm{Ba}}^{2+}$, ${\mathrm{Sr}}^{2+}$, and ${\mathrm{Ca}}^{2+}$.
These cations are precipitated as their respective insoluble carbonates in alkaline medium.
Therefore, Q matches with (4) ${\mathrm{Ba}}^{2+}$.

3. Match for (R): ${\mathrm{NH}}_4\mathrm{OH}$ in the presence of ${\mathrm{NH}}_4\mathrm{Cl}$
This combination represents the group reagent for Group III cations, which includes ${\mathrm{Fe}}^{3+}$, ${\mathrm{Al}}^{3+}$, and ${\mathrm{Cr}}^{3+}$.
The addition of ammonium chloride (${\mathrm{NH}}_4\mathrm{Cl}$) suppresses the ionization of ammonium hydroxide (${\mathrm{NH}}_4\mathrm{OH}$) due to the common ion effect (${\mathrm{NH}}_4^{+}$). This decreases the concentration of hydroxide ions (${\mathrm{OH}}^{-}$) to a level that is only high enough to exceed the low solubility products of Group III metal hydroxides (precipitating $\mathrm{Al}{\left(\mathrm{OH}\right)}_3$) while preventing the premature precipitation of later group hydroxides.
Therefore, R matches with (2) ${\mathrm{Al}}^{3+}$.

4. Match for (S): Passing $H_{2}S$ in the presence of dilute $\mathrm{HCl}$
This is the group reagent for Group II cations, which includes ${\mathrm{Cu}}^{2+}$, ${\mathrm{Pb}}^{2+}$, ${\mathrm{Bi}}^{3+}$, and others.
The presence of hydrochloric acid ($\mathrm{HCl}$) introduces a high concentration of hydrogen ions ($H^{+}$), suppressing the dissociation of the weak acid $H_{2}S$ through the common ion effect. The resulting low concentration of sulfide ions ($S^{2-}$) is only sufficient to precipitate Group II cations because their sulfides have extremely small solubility products.
Therefore, S matches with (1) ${\mathrm{Cu}}^{2+}$.