If `a+b+c=7`, and `a^(2)+b^(2)+c^(2)=33`, then what is the value of `a^(3)+b^(3)+c^(3)-3abc` ?

Options

A. 257
B. 287
C. 343
D. 175

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 + c^3 - 3abc \) given that \( a + b + c = 7 \) and \( a^2 + b^2 + c^2 = 33 \).

### Step 1: Use the identity for \( a^3 + b^3 + c^3 - 3abc \)

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 2: Substitute known values

From the problem, we know:
- \( a + b + c = 7 \)
- \( a^2 + b^2 + c^2 = 33 \)

We need to find \( ab + ac + bc \).

### Step 3: Find \( ab + ac + bc \)

We can use the identity:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]

Substituting the known values:
\[ 7^2 = 33 + 2(ab + ac + bc) \]
\[ 49 = 33 + 2(ab + ac + bc) \]

### Step 4: Solve for \( ab + ac + bc \)

Rearranging the equation:
\[ 49 - 33 = 2(ab + ac + bc) \]
\[ 16 = 2(ab + ac + bc) \]
\[ ab + ac + bc = \frac{16}{2} = 8 \]

### Step 5: Substitute back into the identity

Now we can substitute \( ab + ac + bc \) back into the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
\[ = 7 \left( 33 - 8 \right) \]
\[ = 7 \times 25 \]
\[ = 175 \]

### Final Answer

Thus, the value of \( a^3 + b^3 + c^3 - 3abc \) is \( \boxed{175} \).

---