Select the correct statement for `[M(AB)b_(2)cd]` .

Options

A. a.All geometrical isomers are optically active .
B. b.It has four trans isomer with respect to b
C. c.It has seven geometrical isomers .
D. d.It has three cis and two trans isomers with respect to b .

Explanation

To solve the question regarding the coordination compound `[M(AB)b_(2)cd]`, we need to analyze the structure and the types of isomers it can form. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Ligands
- The compound `[M(AB)b_(2)cd]` consists of a central metal ion `M` coordinated to a bidentate ligand `AB` and two monodentate ligands `b` and `c` and `d`.
- The ligand `AB` is unsymmetrical and bidentate, meaning it can bind to the metal at two sites.

**Hint:** Identify the types of ligands present in the coordination compound and their bonding capabilities.

### Step 2: Draw the Structure
- Start by sketching the structure of the complex. Place the metal `M` at the center.
- Attach the bidentate ligand `AB` to the metal, which will occupy two coordination sites.
- Attach the two `b` ligands and the `c` and `d` ligands to the remaining coordination sites.

**Hint:** Visualizing the structure helps in understanding the spatial arrangement of the ligands around the metal.

### Step 3: Identify Isomerism
- **Cis Isomers:** Since `AB` is bidentate and unsymmetrical, it can lead to different spatial arrangements with the other ligands.
- You can have multiple cis isomers depending on how `b`, `c`, and `d` are arranged around the metal.
- **Trans Isomers:** In the trans configuration, the ligands will be opposite each other, leading to fewer unique arrangements.

**Hint:** Consider how the ligands can be arranged differently to form cis and trans isomers.

### Step 4: Count the Isomers
- For the cis isomers, you can have several arrangements due to the unsymmetrical nature of `AB`. In total, there are 4 possible cis isomers.
- For the trans isomers, you can have 2 distinct arrangements due to the symmetry of the ligands.

**Hint:** Keep track of the different arrangements as you count the isomers.

### Step 5: Determine Optical Activity
- **Cis Isomers:** Since the cis isomers lack a plane of symmetry, they are optically active.
- **Trans Isomers:** The trans isomers have a plane of symmetry and are not optically active.

**Hint:** Recall the definition of optical activity and how symmetry affects it.

### Step 6: Evaluate the Options
- Now, evaluate the given options based on the number of isomers and their optical activity:
- Option A: Incorrect, as trans isomers are not optically active.
- Option B: Incorrect, as there are 4 cis isomers, not 3.
- Option C: Correct, as it states that there are several geometrical isomers.
- Option D: Incorrect, as it misrepresents the number of isomers.

**Hint:** Carefully read each option and match it with your findings.

### Conclusion
The correct statement for the coordination compound `[M(AB)b_(2)cd]` is that it has several geometrical isomers, specifically 4 cis isomers and 2 trans isomers, with the cis isomers being optically active.

### Final Answer
**Option C** is the correct answer.