To solve this problem, we need to understand the group numbers of the given cations based on their qualitative analysis.$\\$The group numbers in qualitative analysis are determined by the precipitation reactions of cations with specific reagents.$\\$Let's analyze each cation:$\\$• $$Al$^{3+}$ belongs to Group III. It forms a precipitate with ammonium hydroxide in the presence of ammonium chloride.$\\$• $$Cu$^{2+}$ belongs to Group II. It forms a precipitate with hydrogen sulfide in an acidic medium.$\\$• $$Ba$^{2+}$ belongs to Group V. It forms a precipitate with ammonium carbonate in the presence of ammonium chloride and ammonium hydroxide.$\\$• $$Co$^{2+}$ belongs to Group II. It forms a precipitate with hydrogen sulfide in an acidic medium.$\\$• $$Mg$^{2+}$ belongs to Group VI. It forms a precipitate with ammonium phosphate in the presence of ammonium chloride and ammonium hydroxide.$\\$Now, let's arrange these cations in increasing order of their group numbers:$\\$1. $$Cu$^{2+}$ (Group II)$\\$2. $$Co$^{2+}$ (Group II)$\\$3. $$Al$^{3+}$ (Group III)$\\$4. $$Ba$^{2+}$ (Group V)$\\$5. $$Mg$^{2+}$ (Group VI)$\\$Therefore, the correct order is: $$B, A, D, C, E$\\$This corresponds to Option 3.