If ` x + y+ z = 9 ` then the value of ` ( x - 4) ^(3) + ( y - 2)^(3) + ( z - 3) ^(3) - 3 ( x - 4) ( y - 2) (z - 3)` is

Options

A. 6
B. 9
C. 0
D. 1

Answer

To solve the expression \( (x - 4)^3 + (y - 2)^3 + (z - 3)^3 - 3(x - 4)(y - 2)(z - 3) \) given that \( x + y + z = 9 \), we can use the identity for the sum of cubes.

### Step-by-Step Solution:

1. **Rewrite the equation**: Start with the given equation \( x + y + z = 9 \). We can express this in terms of \( (x - 4) \), \( (y - 2) \), and \( (z - 3) \):
\[ (x - 4) + (y - 2) + (z - 3) = x + y + z - 9 = 0 \]
Let \( a = x - 4 \), \( b = y - 2 \), and \( c = z - 3 \). Then, we have:
\[ a + b + c = 0 \]

2. **Apply the identity**: We can use the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
Since \( a + b + c = 0 \), we can substitute this into the identity:
\[ a^3 + b^3 + c^3 - 3abc = 0 \cdot (a^2 + b^2 + c^2 - ab - ac - bc) = 0 \]

3. **Conclusion**: Therefore, we find that:
\[ (x - 4)^3 + (y - 2)^3 + (z - 3)^3 - 3(x - 4)(y - 2)(z - 3) = 0 \]

### Final Answer:
The value of the expression is \( 0 \).