Number of geometrical isomers possible for[MABCDEF] are

Options

A. 6
B. 10
C. 15
D. 12

Answer

To determine the number of geometrical isomers possible for the complex [MABCDEF], we can follow these steps:

### Step 1: Identify the Geometry of the Complex
The complex [MABCDEF] has six ligands (A, B, C, D, E, F), which indicates that it adopts an octahedral geometry.

**Hint:** Remember that octahedral complexes can have different arrangements of ligands leading to geometrical isomers.

### Step 2: Understand Geometrical Isomers
Geometrical isomers arise when ligands can occupy different positions relative to each other. In octahedral complexes, we typically consider cis and trans arrangements.

**Hint:** Geometrical isomers can be classified as cis (adjacent ligands) and trans (opposite ligands).

### Step 3: Fix Two Ligands
To find the number of geometrical isomers, we can fix two ligands in specific positions. For example, we can fix ligands A and B at the top and bottom of the octahedron.

**Hint:** Fixing two ligands helps simplify the arrangement of the remaining ligands.

### Step 4: Calculate the Combinations of Remaining Ligands
After fixing two ligands, we have four remaining ligands (C, D, E, F) that can be arranged in various ways. We can calculate the number of ways to choose 2 ligands from the 6 to fix their positions.

The formula for combinations is given by:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
Where \( n \) is the total number of ligands and \( r \) is the number of ligands to choose.

**Hint:** Use the combination formula to determine how many pairs of ligands can be fixed.

### Step 5: Apply the Combination Formula
For our case:
- Total ligands \( n = 6 \)
- Ligands to fix \( r = 2 \)

Calculating \( C(6, 2) \):
\[ C(6, 2) = \frac{6!}{2!(6-2)!} = \frac{6!}{2! \cdot 4!} = \frac{6 \times 5 \times 4!}{2! \times 4!} = \frac{30}{2} = 15 \]

**Hint:** Factorials can simplify the calculation significantly, especially when terms cancel out.

### Step 6: Conclusion
Thus, the total number of geometrical isomers for the complex [MABCDEF] is 15.

**Final Answer:** The correct option is C: 15.