When `a + b + c = (x)/(2) , ` then the value of `((x)/(3 ) - 2a) ^(3) + ((x)/(3) - 2b) ^(3) + ((x)/(3) - 2c) ^(3) - 3 ((x)/(3) - 2a) ((x)/(3) - 2b)((x)/(3) - 2c)` is

Options

A. 0
B. 1
C. 3x
D. 6x

Answer

To solve the problem step by step, let's start with the expression we need to evaluate:

Given:
\[ a + b + c = \frac{x}{2} \]

We need to find the value of:
\[ \left(\frac{x}{3} - 2a\right)^3 + \left(\frac{x}{3} - 2b\right)^3 + \left(\frac{x}{3} - 2c\right)^3 - 3\left(\frac{x}{3} - 2a\right)\left(\frac{x}{3} - 2b\right)\left(\frac{x}{3} - 2c\right) \]

### Step 1: Identify the terms
Let:
\[ A = \frac{x}{3} - 2a, \quad B = \frac{x}{3} - 2b, \quad C = \frac{x}{3} - 2c \]

### Step 2: Use the identity for cubes
We can use the identity:
\[ A^3 + B^3 + C^3 - 3ABC = (A + B + C)(A^2 + B^2 + C^2 - AB - AC - BC) \]

### Step 3: Calculate \(A + B + C\)
Now, calculate \(A + B + C\):
\[ A + B + C = \left(\frac{x}{3} - 2a\right) + \left(\frac{x}{3} - 2b\right) + \left(\frac{x}{3} - 2c\right) \]
\[ = 3 \cdot \frac{x}{3} - 2(a + b + c) = x - 2 \cdot \frac{x}{2} = x - x = 0 \]

### Step 4: Substitute back into the identity
Since \(A + B + C = 0\), we can substitute this into our identity:
\[ A^3 + B^3 + C^3 - 3ABC = 0 \cdot (A^2 + B^2 + C^2 - AB - AC - BC) = 0 \]

### Step 5: Conclusion
Thus, the value of the entire expression simplifies to:
\[ A^3 + B^3 + C^3 - 3ABC = 0 \]

### Final Answer
The value of the expression is:
\[ \boxed{0} \]