If `a + b + c = 10` and `ab + bc + ca = 32` then `a^(3) + b^(3) + c^(3) - 3abc` is equal to :

Options

A. 50
B. 40
C. 60
D. 70

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 + c^3 - 3abc \) given that \( a + b + c = 10 \) and \( ab + bc + ca = 32 \).

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 \( a^2 + b^2 + c^2 \)

We know that:

\[ a + b + c = 10 \]

We can square this equation:

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

Substituting the known values:

\[ 10^2 = a^2 + b^2 + c^2 + 2 \cdot 32 \]

This simplifies to:

\[ 100 = a^2 + b^2 + c^2 + 64 \]

Now, solving for \( a^2 + b^2 + c^2 \):

\[ a^2 + b^2 + c^2 = 100 - 64 = 36 \]

### Step 2: Calculate \( a^2 + b^2 + c^2 - ab - ac - bc \)

We already have \( a^2 + b^2 + c^2 = 36 \) and \( ab + ac + bc = 32 \).

Now we can find:

\[ a^2 + b^2 + c^2 - ab - ac - bc = 36 - 32 = 4 \]

### Step 3: Substitute into the identity

Now we can substitute back into our identity:

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

Substituting the values we found:

\[ a^3 + b^3 + c^3 - 3abc = 10 \cdot 4 = 40 \]

### Conclusion

Thus, the value of \( a^3 + b^3 + c^3 - 3abc \) is \( 40 \).