Given below are two statements :

Statement I : [Co(NH₃)₆]³⁺ is a homoleptic complex whereas [Co(NH₃)₄Cl₂]⁺ is a heteroleptic complex.

Statement II : Complex [Co(NH₃)₆]³⁺ has only one kind of ligands but [Co(NH₃)₄Cl₂]⁺ has more than one kind of ligands.

In the light of the above statements, choose the correct answer from the options given below.

The correct option is: Both Statement I and Statement II are true.

Let us analyze the two statements step-by-step to understand the concepts of homoleptic and heteroleptic complexes:

Analysis of Statement I:
A coordination complex is classified as homoleptic if the central metal atom or ion is bound to only one kind of donor group (ligand). For example, in the complex ${\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_6\right]}^{3+}$, the cobalt ion (${\mathrm{Co}}^{3+}$) is coordinated exclusively to six ammine (${\mathrm{NH}}_3$) ligands. Since all ligands are identical, it is indeed a homoleptic complex.
Conversely, a coordination complex is classified as heteroleptic if the metal is bound to more than one kind of ligand. In the complex ${\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_4{\mathrm{Cl}}_2\right]}^{+}$, the central cobalt ion is bound to two different types of ligands: four ammine (${\mathrm{NH}}_3$) ligands and two chlorido (${\mathrm{Cl}}^{-}$) ligands. Therefore, it is a heteroleptic complex.
Thus, Statement I is true.

Analysis of Statement II:
Statement II defines the underlying reason for the classifications mentioned in Statement I. It states that ${\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_6\right]}^{3+}$ has only one kind of ligand (${\mathrm{NH}}_3$), while ${\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_4{\mathrm{Cl}}_2\right]}^{+}$ has more than one kind of ligand (${\mathrm{NH}}_3$ and ${\mathrm{Cl}}^{-}$). This matches the definitions of homoleptic and heteroleptic complexes perfectly.
Thus, Statement II is also true.

Since both Statement I and Statement II are correct, the correct choice is that both statements are true.