How many of the following complexes are optically inactive?
(i) trans-`[Co(NH_3)_4I_2]^+`
(ii) cis-`[Co(NH_3)_2(en)_2]^(3+)`
(iii) trans-`[Co(NH_3)_2(en)_2]^(3+)`

Explanation

To determine how many of the given complexes are optically inactive, we need to analyze each complex for symmetry elements. A complex is optically inactive if it possesses a plane of symmetry, a center of symmetry, or an alternate axis of symmetry.

### Step 1: Analyze the first complex - trans-`[Co(NH₃)₄I₂]⁺`

1. **Identify the geometry**: The complex is octahedral due to the coordination number of 6 (4 NH₃ and 2 I).
2. **Determine the arrangement**: In the trans configuration, the two I ligands are opposite each other.
3. **Check for symmetry**: The presence of two identical ligands (I) opposite each other creates a plane of symmetry through the center of the complex.
4. **Conclusion**: Since it has a plane of symmetry, trans-`[Co(NH₃)₄I₂]⁺` is optically inactive.

### Step 2: Analyze the second complex - cis-`[Co(NH₃)₂(en)₂]³⁺`

1. **Identify the geometry**: The complex is also octahedral.
2. **Determine the arrangement**: In the cis configuration, the two NH₃ ligands are adjacent to each other, and the two ethylenediamine (en) ligands are also adjacent.
3. **Check for symmetry**: This arrangement does not have a plane of symmetry or a center of symmetry due to the different ligands (NH₃ and en) and their positions.
4. **Conclusion**: Since it lacks symmetry elements, cis-`[Co(NH₃)₂(en)₂]³⁺` is optically active.

### Step 3: Analyze the third complex - trans-`[Co(NH₃)₂(en)₂]³⁺`

1. **Identify the geometry**: This complex is octahedral as well.
2. **Determine the arrangement**: In the trans configuration, the two NH₃ ligands are opposite each other, and the two en ligands are also opposite each other.
3. **Check for symmetry**: Similar to the first complex, the presence of identical ligands (NH₃) opposite each other creates a plane of symmetry.
4. **Conclusion**: Since it has a plane of symmetry, trans-`[Co(NH₃)₂(en)₂]³⁺` is optically inactive.

### Final Summary

- **Optically inactive complexes**:
- trans-`[Co(NH₃)₄I₂]⁺`
- trans-`[Co(NH₃)₂(en)₂]³⁺`

- **Optically active complex**:
- cis-`[Co(NH₃)₂(en)₂]³⁺`

Thus, **2 out of the 3 complexes are optically inactive**.