`(113^(2) + 115^(2) + 117^(2) - 113 xx 115 - 115 xx 117 - 117 xx 113)` is equal to

Options

A. 0
B. 4
C. 8
D. 12

Answer

To solve the expression \( (113^2 + 115^2 + 117^2 - 113 \times 115 - 115 \times 117 - 117 \times 113) \), we can follow these steps:

### Step 1: Identify the variables
Let:
- \( a = 113 \)
- \( b = 115 \)
- \( c = 117 \)

### Step 2: Rewrite the expression
The expression can be rewritten using the variables:
\[ a^2 + b^2 + c^2 - ab - bc - ca \]

### Step 3: Use the identity
We can use the identity:
\[ a^2 + b^2 + c^2 - ab - bc - ca = \frac{1}{2} \left( (a-b)^2 + (b-c)^2 + (c-a)^2 \right) \]

### Step 4: Calculate each term
Now we calculate each squared difference:
- \( (a-b)^2 = (113 - 115)^2 = (-2)^2 = 4 \)
- \( (b-c)^2 = (115 - 117)^2 = (-2)^2 = 4 \)
- \( (c-a)^2 = (117 - 113)^2 = (4)^2 = 16 \)

### Step 5: Substitute back into the identity
Now substitute these values back into the identity:
\[ \frac{1}{2} \left( 4 + 4 + 16 \right) = \frac{1}{2} \times 24 = 12 \]

### Final Answer
Thus, the value of the original expression is:
\[ \boxed{12} \]