A metal complex of coordination number six having three different types of ligands a, b and c of composition `Ma_(2)b_(2)c_(2)` can exist in several geometrical isomeric forms, the total number of such isomers is:

Options

A. 3
B. 5
C. 7
D. 9

Answer

To determine the total number of geometrical isomers for the metal complex \( Ma_2b_2c_2 \) with a coordination number of six and three different types of ligands (a, b, and c), we can follow these steps:

### Step 1: Understand the Coordination Geometry
The coordination number of six typically leads to an octahedral geometry. In this geometry, the ligands can be arranged in different ways around the central metal atom.

### Step 2: Identify the Types of Ligands
We have three types of ligands: A, B, and C. Each type appears twice in the formula \( Ma_2b_2c_2 \).

### Step 3: Determine Possible Geometrical Isomers
In an octahedral complex with the given ligand arrangement, we can have different geometrical isomers based on the positioning of the ligands. The arrangements can be categorized into two main types:
1. **Facial (fac) Isomers**: Where two of the same type of ligands occupy adjacent positions.
2. **Meridional (mer) Isomers**: Where two of the same type of ligands occupy positions that are opposite each other.

### Step 4: Count the Isomers
For the complex \( Ma_2b_2c_2 \):
- **Facial Isomers**:
- One possible arrangement is having all A's adjacent to each other, which can be arranged as \( AABBC \) or \( AABCC \) or \( AACB \) etc.
- This can lead to several arrangements where A's are together, and B's and C's are placed accordingly.

- **Meridional Isomers**:
- The arrangement where A's are opposite each other, and B's and C's are placed in the remaining positions.

After considering all possible arrangements, we find that there are a total of **5 distinct geometrical isomers** for the complex \( Ma_2b_2c_2 \).

### Final Answer
The total number of geometrical isomers for the complex \( Ma_2b_2c_2 \) is **5**.

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