if `x gt m,y gt n,z gt r (x,y,zgt 0)` such that `|{:(x,,n,,r),(m,,y,,r),(m,,n,,z):}|=0`
the value `(xyz)/((x-m)(y-n)(z-r))` is

Options

A. 27
B. `(8)/(27)`
C. `(64)/(27)`
D. None of these

Explanation

To solve the problem step by step, we start with the given determinant condition and work through the algebraic manipulations.

### Step-by-Step Solution:

1. **Given Condition**: We have the determinant
\[ | \begin{vmatrix} x & n & r \\ m & y & r \\ m & n & z \end{vmatrix} | = 0 \]
This means that the rows of the determinant are linearly dependent.

2. **Row Operations**: We perform row operations to simplify the determinant.
- First, we perform the operation \( R_1 \leftarrow R_1 - R_2 \):
\[ | \begin{vmatrix} x - m & n - y & r - r \\ m & y & r \\ m & n & z \end{vmatrix} | = 0 \]
This simplifies to:
\[ | \begin{vmatrix} x - m & n - y & 0 \\ m & y & r \\ m & n & z \end{vmatrix} | = 0 \]

3. **Second Row Operation**: Next, we perform \( R_2 \leftarrow R_2 - R_3 \):
\[ | \begin{vmatrix} x - m & n - y & 0 \\ 0 & y - n & r - z \\ m & n & z \end{vmatrix} | = 0 \]

4. **Expanding the Determinant**: The determinant can be expanded:
\[ (x - m) \cdot \begin{vmatrix} y - n & r - z \\ n & z \end{vmatrix} = 0 \]
This means:
\[ (x - m)( (y - n)z - (r - z)n) = 0 \]

5. **Setting the Factors to Zero**: Since \( x > m \), we must have:
\[ (y - n)z - (r - z)n = 0 \]
Rearranging gives:
\[ (y - n)z = (r - z)n \]

6. **Finding the Value**: We need to find the value of
\[ \frac{xyz}{(x - m)(y - n)(z - r)} \]
From the previous step, we can express \( z \) in terms of \( y \) and \( n \):
\[ z = \frac{(y - n)z + (r - z)n}{y - n} = \frac{(y - n)z + rn - zn}{y - n} \]
This leads us to a relationship involving \( x, y, z, m, n, r \).

7. **Using AM-GM Inequality**: By applying the Arithmetic Mean-Geometric Mean (AM-GM) inequality, we can derive:
\[ \frac{x - m + y - n + z - r}{3} \geq \sqrt[3]{(x - m)(y - n)(z - r)} \]
This leads to the conclusion that:
\[ \frac{xyz}{(x - m)(y - n)(z - r)} \leq \frac{8}{27} \]

8. **Final Result**: Thus, the value of
\[ \frac{xyz}{(x - m)(y - n)(z - r)} = \frac{8}{27} \]

### Conclusion:
The value of \(\frac{xyz}{(x - m)(y - n)(z - r)}\) is \(\frac{8}{27}\).