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

Options

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

Answer

To solve the problem, we need to determine how many different ways we can decompose a third-order determinant based on the structure of its columns. Let's break it down step by step.

### Step 1: Understanding the Structure of the Determinant
We have a third-order determinant (3x3) with the following structure:
- The first column consists of elements that are sums of two terms:
- \( a_1 + b_1 \)
- \( a_2 + b_2 \)
- \( a_3 + b_3 \)

- The second column consists of elements that are sums of three terms:
- \( c_1 + d_1 + e_1 \)
- \( c_2 + d_2 + e_2 \)
- \( c_3 + d_3 + e_3 \)

- The third column consists of elements that are sums of four terms:
- \( f_1 + g_1 + h_1 + i_1 \)
- \( f_2 + g_2 + h_2 + i_2 \)
- \( f_3 + g_3 + h_3 + i_3 \)

### Step 2: Decomposing the First Column
The first column can be decomposed into two separate determinants:
1. One determinant where the first column consists of \( a_1, a_2, a_3 \)
2. Another determinant where the first column consists of \( b_1, b_2, b_3 \)

Thus, we can decompose the determinant in **2 ways** based on the first column.

### Step 3: Decomposing the Second Column
Next, we consider the second column, which can be decomposed into three separate determinants:
1. One determinant where the second column consists of \( c_1, c_2, c_3 \)
2. Another determinant where the second column consists of \( d_1, d_2, d_3 \)
3. A third determinant where the second column consists of \( e_1, e_2, e_3 \)

Thus, we can decompose the determinant in **3 ways** based on the second column.

### Step 4: Decomposing the Third Column
Finally, we look at the third column, which can be decomposed into four separate determinants:
1. One determinant where the third column consists of \( f_1, f_2, f_3 \)
2. Another determinant where the third column consists of \( g_1, g_2, g_3 \)
3. A third determinant where the third column consists of \( h_1, h_2, h_3 \)
4. A fourth determinant where the third column consists of \( i_1, i_2, i_3 \)

Thus, we can decompose the determinant in **4 ways** based on the third column.

### Step 5: Calculating the Total Number of Determinants
To find the total number of determinants we can form, we multiply the number of ways we can decompose each column:
\[ n = 2 \times 3 \times 4 = 24 \]

### Conclusion
The value of \( n \) is **24**.