What is the value equation ` a^(3) + b^(3) + c^(3) - 3abc and If a^(2) + b^(2) + c^(2) = ab + bc + ca + 4 and` `
` `a + b + c = 4` ?

Options

A. 0
B. 1
C. 16
D. 256

Answer

To solve the equation \( a^3 + b^3 + c^3 - 3abc \) given the conditions \( a^2 + b^2 + c^2 = ab + bc + ca + 4 \) and \( a + b + c = 4 \), we can use a known algebraic identity.

### Step-by-Step Solution:

1. **Use the identity for the sum of cubes**:
The expression \( a^3 + b^3 + c^3 - 3abc \) can be rewritten using the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

2. **Substitute the known value**:
We know from the problem statement that \( a + b + c = 4 \). Thus, we can substitute this into the equation:
\[ a^3 + b^3 + c^3 - 3abc = 4(a^2 + b^2 + c^2 - ab - ac - bc) \]

3. **Simplify the expression inside the parentheses**:
We also have the equation \( a^2 + b^2 + c^2 = ab + ac + bc + 4 \). We can rearrange this to find \( a^2 + b^2 + c^2 - ab - ac - bc \):
\[ a^2 + b^2 + c^2 - ab - ac - bc = 4 \]

4. **Substitute back into the equation**:
Now we can substitute this result back into our earlier expression:
\[ a^3 + b^3 + c^3 - 3abc = 4 \times 4 = 16 \]

5. **Final answer**:
Therefore, the value of the expression \( a^3 + b^3 + c^3 - 3abc \) is:
\[ \boxed{16} \]