If p+q+r=9 then `(3-p)^(3)+(3-q)6(3)+(3-r)^(3)` is :

Options

A. 3(3-p)(3-q)(3-r)
B. 0
C. 1
D. `-3(3-p)(3-q)(3-r)`

Answer

To solve the problem, we need to evaluate the expression \((3-p)^3 + (3-q)^3 + (3-r)^3\) given that \(p + q + r = 9\).

### Step-by-Step Solution:

1. **Identify the Expression**:
We have the expression \((3-p)^3 + (3-q)^3 + (3-r)^3\).

2. **Let \(x = 3 - p\), \(y = 3 - q\), and \(z = 3 - r\)**:
This gives us:
\[ x + y + z = (3 - p) + (3 - q) + (3 - r) = 9 - (p + q + r) = 9 - 9 = 0 \]

3. **Use the Identity for Sums of Cubes**:
We know that if \(x + y + z = 0\), then:
\[ x^3 + y^3 + z^3 = 3xyz \]
Therefore, we can rewrite our expression as:
\[ (3-p)^3 + (3-q)^3 + (3-r)^3 = 3(3-p)(3-q)(3-r) \]

4. **Calculate \(xyz\)**:
We need to find \(xyz\):
\[ xyz = (3-p)(3-q)(3-r) \]
Substituting \(x = 3 - p\), \(y = 3 - q\), and \(z = 3 - r\):
\[ xyz = (3-p)(3-q)(3-r) \]

5. **Expand the Product**:
To expand \((3-p)(3-q)(3-r)\):
\[ = 27 - 9(p + q + r) + 3(pq + qr + rp) - pqr \]
Since \(p + q + r = 9\), we have:
\[ = 27 - 9 \times 9 + 3(pq + qr + rp) - pqr \]
\[ = 27 - 81 + 3(pq + qr + rp) - pqr \]
\[ = -54 + 3(pq + qr + rp) - pqr \]

6. **Final Expression**:
Therefore, the original expression becomes:
\[ (3-p)^3 + (3-q)^3 + (3-r)^3 = 3xyz = 3(-54 + 3(pq + qr + rp) - pqr) \]

### Conclusion:
The value of \((3-p)^3 + (3-q)^3 + (3-r)^3\) is \(3(-54 + 3(pq + qr + rp) - pqr)\).