How many geometrical isomers are possible for `MA_(2)B_(2)C_(2)` ?

Answer

To determine how many geometrical isomers are possible for the coordination compound \( MA_2B_2C_2 \), we will analyze the arrangement of the ligands around the central metal atom \( M \).

### Step-by-Step Solution:

1. **Identify the Coordination Environment:**
The complex \( MA_2B_2C_2 \) consists of one metal ion \( M \) coordinated to two \( A \) ligands, two \( B \) ligands, and two \( C \) ligands.

2. **Understand Geometrical Isomerism:**
Geometrical isomers arise due to different spatial arrangements of the ligands around the central metal ion. In this case, we can have cis and trans arrangements for the ligands.

3. **Consider Different Arrangements:**
- **Cis Arrangement:** In a cis arrangement, the same type of ligands are adjacent to each other. For example, if we have \( A \) ligands together, we can have:
- \( A A \) cis, \( B B \) cis, \( C C \) cis.
- **Trans Arrangement:** In a trans arrangement, the same type of ligands are opposite each other. For example:
- \( A A \) trans, \( B B \) trans, \( C C \) trans.

4. **Count Possible Isomers:**
- **Case 1:** \( A \) ligands are cis, \( B \) and \( C \) can be either cis or trans.
- **Case 2:** \( B \) ligands are cis, \( A \) and \( C \) can be either cis or trans.
- **Case 3:** \( C \) ligands are cis, \( A \) and \( B \) can be either cis or trans.
- **All Ligands Cis:** All \( A \), \( B \), and \( C \) are cis.
- **All Ligands Trans:** All \( A \), \( B \), and \( C \) are trans.

5. **List the Unique Isomers:**
- \( A A \) cis, \( B B \) cis, \( C C \) trans (1)
- \( A A \) cis, \( B B \) trans, \( C C \) cis (2)
- \( A A \) trans, \( B B \) cis, \( C C \) cis (3)
- \( A A \) cis, \( B B \) cis, \( C C \) cis (4)
- \( A A \) trans, \( B B \) trans, \( C C \) trans (5)

6. **Final Count:**
After analyzing all possible arrangements, we find that there are a total of **5 geometrical isomers** for the complex \( MA_2B_2C_2 \).

### Final Answer:
The total number of geometrical isomers possible for \( MA_2B_2C_2 \) is **5**.