In a third order determinant, each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of the third column consists of sum of four terms. Then it can be decomposed into n determinants, where n has the value

Options

A. 1
B. 9
C. 16
D. 24

Answer

To solve the problem, we need to determine how many smaller determinants a third-order determinant can be decomposed into, given that each element in the first column consists of a sum of two terms, each element in the second column consists of a sum of three terms, and each element in the third column consists of a sum of four terms.

### Step-by-Step Solution:

1. **Understanding the Structure of the Determinant:**
We are dealing with a 3x3 determinant (third order). Each element in the columns has a specific structure:
- First column: Each element is a sum of 2 terms.
- Second column: Each element is a sum of 3 terms.
- Third column: Each element is a sum of 4 terms.

2. **Decomposing Each Column:**
- The first column can be decomposed into 2 determinants because each element consists of 2 terms.
- The second column can be decomposed into 3 determinants because each element consists of 3 terms.
- The third column can be decomposed into 4 determinants because each element consists of 4 terms.

3. **Using the Formula for Decomposition:**
We can use the formula that states if the elements of a determinant of order 3 consist of \( m \), \( n \), and \( p \) terms respectively, then the determinant can be expressed as:
\[ m \times n \times p \]
In our case:
- \( m = 2 \) (for the first column)
- \( n = 3 \) (for the second column)
- \( p = 4 \) (for the third column)

4. **Calculating the Total Number of Determinants:**
Now, we can calculate the total number of determinants:
\[ \text{Total determinants} = 2 \times 3 \times 4 = 24 \]

5. **Conclusion:**
Therefore, the original determinant can be decomposed into 24 smaller determinants.

### Final Answer:
The value of \( n \) is 24.

---