The value of `|(.^(10)C_(4).^(10)C_(5).^(11)C_(m)),(.^(11)C_(6).^(11)C_(7).^(12)C_(m+2)),(.^(12)C_(8).^(12)C_(9).^(13)C_(m+4))|` is equal to zero when `m` is

Options

A. `6`
B. `4`
C. `5`
D. `7`

Explanation

To solve the problem, we need to find the value of \( m \) such that the determinant

\[ \left| \begin{array}{ccc} ^{10}C_{4} & ^{10}C_{5} & ^{11}C_{m} \\ ^{11}C_{6} & ^{11}C_{7} & ^{12}C_{m+2} \\ ^{12}C_{8} & ^{12}C_{9} & ^{13}C_{m+4} \end{array} \right| = 0 \]

### Step 1: Understand the properties of determinants
We know that if two columns of a determinant are identical, then the value of the determinant is zero. We will manipulate the columns to see if we can make two columns identical.

### Step 2: Modify the second column
We will replace the second column with the sum of the first column and the second column:

\[ \text{New Column 2} = \text{Column 1} + \text{Column 2} \]

So, the new determinant becomes:

\[ \left| \begin{array}{ccc} ^{10}C_{4} & ^{10}C_{4} + ^{10}C_{5} & ^{11}C_{m} \\ ^{11}C_{6} & ^{11}C_{6} + ^{11}C_{7} & ^{12}C_{m+2} \\ ^{12}C_{8} & ^{12}C_{8} + ^{12}C_{9} & ^{13}C_{m+4} \end{array} \right| \]

### Step 3: Apply the property of combinations
Using the property \( ^nC_r + ^nC_{r+1} = ^{n+1}C_{r+1} \):

- For the first row:
\[ ^{10}C_{4} + ^{10}C_{5} = ^{11}C_{5} \]
- For the second row:
\[ ^{11}C_{6} + ^{11}C_{7} = ^{12}C_{7} \]
- For the third row:
\[ ^{12}C_{8} + ^{12}C_{9} = ^{13}C_{9} \]

Thus, the determinant simplifies to:

\[ \left| \begin{array}{ccc} ^{10}C_{4} & ^{11}C_{5} & ^{11}C_{m} \\ ^{11}C_{6} & ^{12}C_{7} & ^{12}C_{m+2} \\ ^{12}C_{8} & ^{13}C_{9} & ^{13}C_{m+4} \end{array} \right| \]

### Step 4: Set two columns equal
Now we want the second and third columns to be equal to make the determinant zero:

\[ ^{11}C_{5} = ^{11}C_{m} \quad \text{and} \quad ^{12}C_{7} = ^{12}C_{m+2} \]

### Step 5: Solve for \( m \)
From \( ^{11}C_{5} = ^{11}C_{m} \), we have:

\[ m = 5 \]

From \( ^{12}C_{7} = ^{12}C_{m+2} \):

\[ m + 2 = 7 \implies m = 5 \]

### Conclusion
Thus, the value of \( m \) that makes the determinant equal to zero is:

\[ \boxed{5} \]