Write the sum of geometrical isomers in `[Ma_(2)b_(2)c_(2)]` complex and stereoisomers in `[M(AB)_(3)]` complex .

Explanation

To solve the question regarding the sum of geometrical isomers in the complex \([Ma_2b_2c_2]\) and stereoisomers in the complex \([M(AB)_3]\), we will break it down into two parts.

### Step 1: Determine the geometrical isomers of \([Ma_2b_2c_2]\)

1. **Identify the ligands**: We have three different monodentate ligands: A, B, and C.
2. **Count the possible arrangements**:
- **Cis Isomers**:
- One possible arrangement is where all A's are adjacent (cis), and B's and C's can also be adjacent. This gives us the first isomer: **AA, BB, CC** (cis).
- **Trans Isomers**:
- The second arrangement can have A's in trans positions, while B's and C's are in cis positions. This gives us the second isomer.
- We can also have B's in cis and A's and C's in trans, giving us the third isomer.
- Another arrangement can have C's in cis and A's and B's in trans, giving us the fourth isomer.
- **All Trans Isomer**:
- Finally, we can have all ligands in trans positions, giving us the fifth isomer.

3. **Total geometrical isomers**: After considering all arrangements, we find that there are a total of **5 geometrical isomers**.

### Step 2: Determine the stereoisomers of \([M(AB)_3]\)

1. **Identify the ligand**: Here, AB is a bidentate ligand, meaning it can attach to the metal at two points.
2. **Count the possible arrangements**:
- The complex can exist in both **cis** and **trans** forms.
- Each of these forms can also exhibit optical activity because the arrangement of the bidentate ligands can create non-superimposable mirror images.
3. **Total stereoisomers**:
- There are **2 forms (cis and trans)**, and since both can be optically active, we multiply by 2, giving us a total of **4 stereoisomers**.

### Final Calculation

Now, we sum the geometrical isomers and stereoisomers:
- Geometrical isomers: 5
- Stereoisomers: 4

Thus, the total sum is:
\[ 5 + 4 = 9 \]

### Final Answer
The sum of geometrical isomers in \([Ma_2b_2c_2]\) and stereoisomers in \([M(AB)_3]\) is **9**.