If `a+b+c=8`, then value of `(a-4)^(3) +(b-3)^(3) +(c-1)^(3)-3(a-4) (b-3) (c-1)` is

Options

A. 2
B. 4
C. 1
D. 0

Answer

To solve the expression \( (a-4)^3 + (b-3)^3 + (c-1)^3 - 3(a-4)(b-3)(c-1) \) given that \( a + b + c = 8 \), we can use a strategic approach by substituting specific values for \( a \), \( b \), and \( c \) that satisfy the equation.

### Step-by-Step Solution:

1. **Choose Values for \( a \), \( b \), and \( c \)**:
Since \( a + b + c = 8 \), we can choose \( a = 4 \), \( b = 3 \), and \( c = 1 \). This satisfies the equation because \( 4 + 3 + 1 = 8 \).

2. **Substitute the Values into the Expression**:
Now, we substitute these values into the expression:
\[ (4-4)^3 + (3-3)^3 + (1-1)^3 - 3(4-4)(3-3)(1-1) \]

3. **Calculate Each Term**:
- First term: \( (4-4)^3 = 0^3 = 0 \)
- Second term: \( (3-3)^3 = 0^3 = 0 \)
- Third term: \( (1-1)^3 = 0^3 = 0 \)
- Fourth term: \( 3(4-4)(3-3)(1-1) = 3 \cdot 0 \cdot 0 \cdot 0 = 0 \)

4. **Combine the Results**:
Now, we combine all the results:
\[ 0 + 0 + 0 - 0 = 0 \]

5. **Final Result**:
Therefore, the value of the expression is \( 0 \).

### Final Answer:
\[ \text{The value of } (a-4)^3 + (b-3)^3 + (c-1)^3 - 3(a-4)(b-3)(c-1) \text{ is } 0. \]