If `a + b + c = 13` and `ab + bc + ca = 54`, then `a^(3) + b^(3) + c^(3) - 3abc` is equal to :

Options

A. A)793
B. B)273
C. C)91
D. D)182

Answer

To solve the problem step by step, we will use the given equations and the formula for \( a^3 + b^3 + c^3 - 3abc \).

### Step 1: Write down the given equations
We have:
1. \( a + b + c = 13 \)
2. \( ab + bc + ca = 54 \)

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

### Step 3: Substitute the known values into the formula
First, we calculate \( (a + b + c)^2 \):
\[ (a + b + c)^2 = 13^2 = 169 \]

Now substitute into the formula:
\[ a^3 + b^3 + c^3 - 3abc = 13 \left( 169 - 3 \times 54 \right) \]

### Step 4: Calculate \( 3 \times 54 \)
\[ 3 \times 54 = 162 \]

### Step 5: Substitute back into the equation
Now we substitute this value back into our equation:
\[ a^3 + b^3 + c^3 - 3abc = 13 \left( 169 - 162 \right) \]

### Step 6: Simplify the expression
\[ 169 - 162 = 7 \]
So we have:
\[ a^3 + b^3 + c^3 - 3abc = 13 \times 7 \]

### Step 7: Calculate \( 13 \times 7 \)
\[ 13 \times 7 = 91 \]

### Final Answer
Thus, the value of \( a^3 + b^3 + c^3 - 3abc \) is \( \boxed{91} \).

---