If `a+b + c = 5` and `a^(2) + b^(2) + c^(2) = 33`, then what is the value of `a^(3) + b^(3) + c^(3) - 3abc` ?

Options

A. 195
B. 180
C. 192
D. 185

Answer

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

1. \( a + b + c = 5 \)
2. \( a^2 + b^2 + c^2 = 33 \)

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 \( ab + ac + bc \)

First, we need to find \( ab + ac + bc \). We can use the square of the sum of \( a, b, \) and \( c \):

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

Substituting the known values:

\[ 5^2 = 33 + 2(ab + ac + bc) \]

This simplifies to:

\[ 25 = 33 + 2(ab + ac + bc) \]

### Step 2: Solve for \( ab + ac + bc \)

Rearranging the equation gives:

\[ 2(ab + ac + bc) = 25 - 33 \]
\[ 2(ab + ac + bc) = -8 \]
\[ ab + ac + bc = -4 \]

### Step 3: Substitute values into the identity

Now we can substitute the values into the identity:

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

Substituting the known values:

\[ = 5 \left( 33 - (-4) \right) \]
\[ = 5 \left( 33 + 4 \right) \]
\[ = 5 \times 37 \]
\[ = 185 \]

### Final Answer

Thus, the value of \( a^3 + b^3 + c^3 - 3abc \) is \( \boxed{185} \).