If `a = 34, b =c = 33,` then the value of `a ^(3) + b ^(3) + c ^(3)- 3 abc` is :

Options

A. A)0
B. B)111
C. C)50
D. D)100

Answer

To solve the problem, we need to calculate the expression \( a^3 + b^3 + c^3 - 3abc \) given that \( a = 34 \), \( b = 33 \), and \( c = 33 \).

### Step-by-step Solution:

1. **Identify the Values**:
- Given: \( a = 34 \), \( b = 33 \), \( c = 33 \)

2. **Calculate \( a + b + c \)**:
\[ a + b + c = 34 + 33 + 33 = 100 \]

3. **Calculate \( a - b \)**:
\[ a - b = 34 - 33 = 1 \]

4. **Calculate \( b - c \)**:
\[ b - c = 33 - 33 = 0 \]

5. **Calculate \( c - a \)**:
\[ c - a = 33 - 34 = -1 \]

6. **Square the Differences**:
- \( (a - b)^2 = 1^2 = 1 \)
- \( (b - c)^2 = 0^2 = 0 \)
- \( (c - a)^2 = (-1)^2 = 1 \)

7. **Sum of the Squares**:
\[ (a - b)^2 + (b - c)^2 + (c - a)^2 = 1 + 0 + 1 = 2 \]

8. **Use the Formula**:
The formula for \( a^3 + b^3 + c^3 - 3abc \) is:
\[ a^3 + b^3 + c^3 - 3abc = \frac{1}{2} (a + b + c) \left( (a - b)^2 + (b - c)^2 + (c - a)^2 \right) \]
Substituting the values we calculated:
\[ a^3 + b^3 + c^3 - 3abc = \frac{1}{2} \times 100 \times 2 \]

9. **Calculate the Final Value**:
\[ = \frac{1}{2} \times 100 \times 2 = 100 \]

### Final Answer:
The value of \( a^3 + b^3 + c^3 - 3abc \) is \( 100 \).

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