If `a + b + c = 2, a^(2) + b^(2) + c^(2) = 26`, then the value of `a^(3) + b^(3) + c^(3) - 3abc` is :

Options

A. A)71
B. B)74
C. C)69
D. D)78

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 + c^3 - 3abc \) given the equations:

1. \( a + b + c = 2 \)
2. \( a^2 + b^2 + c^2 = 26 \)

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 1: Calculate \( ab + ac + bc \)

From the first equation, we can square it:

\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]

Substituting the known values:

\[ 2^2 = 26 + 2(ab + ac + bc) \]

This simplifies to:

\[ 4 = 26 + 2(ab + ac + bc) \]

Rearranging gives:

\[ 2(ab + ac + bc) = 4 - 26 \]

\[ 2(ab + ac + bc) = -22 \]

Dividing by 2:

\[ ab + ac + bc = -11 \]

### Step 2: Substitute values into the identity

Now we have:

- \( a + b + c = 2 \)
- \( a^2 + b^2 + c^2 = 26 \)
- \( ab + ac + bc = -11 \)

Substituting these into the identity:

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

Calculating \( a^2 + b^2 + c^2 - ab - ac - bc \):

\[ a^2 + b^2 + c^2 - ab - ac - bc = 26 - (-11) = 26 + 11 = 37 \]

Now substituting back into the identity:

\[ a^3 + b^3 + c^3 - 3abc = 2 \times 37 = 74 \]

### Final Answer

Thus, the value of \( a^3 + b^3 + c^3 - 3abc \) is:

\[ \boxed{74} \]