If a +b+c=7 and `a^(3)+b^(3)+c^(3)-3abc=175` ,then what is the value of ab +bc +ca ?

Options

A. 9
B. 7
C. 10
D. 8

Answer

To solve the problem, we will use the given equations and the formula relating the sums of cubes to the sums of the variables and their products.

### Step-by-Step Solution:

1. **Given Equations**:
- We know that \( a + b + c = 7 \).
- We also have \( a^3 + b^3 + c^3 - 3abc = 175 \).

2. **Use the Formula**:
- The formula we will use is:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c) \left( (a + b + c)^2 - 3(ab + ac + bc) \right) \]

3. **Substituting Known Values**:
- Substitute \( a + b + c = 7 \) into the formula:
\[ a^3 + b^3 + c^3 - 3abc = 7 \left( 7^2 - 3(ab + ac + bc) \right) \]
- This simplifies to:
\[ a^3 + b^3 + c^3 - 3abc = 7 \left( 49 - 3(ab + ac + bc) \right) \]

4. **Set the Equation**:
- We know \( a^3 + b^3 + c^3 - 3abc = 175 \), so we can set up the equation:
\[ 7 \left( 49 - 3(ab + ac + bc) \right) = 175 \]

5. **Divide Both Sides by 7**:
- Dividing both sides by 7 gives:
\[ 49 - 3(ab + ac + bc) = 25 \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ 49 - 25 = 3(ab + ac + bc) \]
- Therefore:
\[ 24 = 3(ab + ac + bc) \]

7. **Solving for \( ab + ac + bc \)**:
- Divide both sides by 3:
\[ ab + ac + bc = \frac{24}{3} = 8 \]

### Final Answer:
The value of \( ab + ac + bc \) is \( 8 \).

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