If `a = 2017, b =2016 and c = 2015,` then what is the value of `a ^(2) + b ^(2) + c ^(2) - ab - bc - ca` ?

Options

A. `-2`
B. 0
C. 3
D. 4

Answer

To solve the expression \( a^2 + b^2 + c^2 - ab - bc - ca \) given \( a = 2017 \), \( b = 2016 \), and \( c = 2015 \), we can follow these steps:

### Step 1: Substitute the values of a, b, and c
We start by substituting the given values into the expression:
\[ a^2 + b^2 + c^2 - ab - bc - ca = 2017^2 + 2016^2 + 2015^2 - (2017 \cdot 2016) - (2016 \cdot 2015) - (2015 \cdot 2017) \]

### Step 2: Calculate \( a^2, b^2, c^2 \)
Now, we calculate each square:
\[ 2017^2 = 4068289 \]
\[ 2016^2 = 4064256 \]
\[ 2015^2 = 4060225 \]

### Step 3: Sum the squares
Next, we sum these squares:
\[ 4068289 + 4064256 + 4060225 = 12182770 \]

### Step 4: Calculate the products \( ab, bc, ca \)
Now, we calculate the products:
\[ ab = 2017 \cdot 2016 = 4066272 \]
\[ bc = 2016 \cdot 2015 = 4062240 \]
\[ ca = 2015 \cdot 2017 = 4068285 \]

### Step 5: Sum the products
Next, we sum these products:
\[ 4066272 + 4062240 + 4068285 = 12126897 \]

### Step 6: Substitute back into the expression
Now, we substitute back into the expression:
\[ 12182770 - 12126897 = 55873 \]

### Step 7: Final Result
Thus, the value of the expression \( a^2 + b^2 + c^2 - ab - bc - ca \) is:
\[ \boxed{55873} \]

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