If a = 7, b = 5 and c = 3 ,then the value of `a^(2) + b^(2) + c^(2) - ab - bc - ca ` is

Options

A. A)12
B. B)`-12`
C. C)0
D. D)8

Answer

To find the value of the expression \( a^2 + b^2 + c^2 - ab - bc - ca \) given \( a = 7 \), \( b = 5 \), and \( c = 3 \), we will follow these steps:

### Step 1: Substitute the values of a, b, and c into the expression
We start by substituting the given values into the expression:

\[ a^2 + b^2 + c^2 - ab - bc - ca = 7^2 + 5^2 + 3^2 - (7 \cdot 5) - (5 \cdot 3) - (3 \cdot 7) \]

### Step 2: Calculate each term
Now we calculate each term in the expression:

1. Calculate \( a^2 \):
\[ 7^2 = 49 \]

2. Calculate \( b^2 \):
\[ 5^2 = 25 \]

3. Calculate \( c^2 \):
\[ 3^2 = 9 \]

4. Calculate \( ab \):
\[ 7 \cdot 5 = 35 \]

5. Calculate \( bc \):
\[ 5 \cdot 3 = 15 \]

6. Calculate \( ca \):
\[ 3 \cdot 7 = 21 \]

### Step 3: Substitute the calculated values back into the expression
Now we substitute these calculated values back into the expression:

\[ 49 + 25 + 9 - 35 - 15 - 21 \]

### Step 4: Simplify the expression
Now we simplify the expression step by step:

1. First, add \( 49 + 25 + 9 \):
\[ 49 + 25 = 74 \]
\[ 74 + 9 = 83 \]

2. Now subtract \( 35 + 15 + 21 \):
\[ 35 + 15 = 50 \]
\[ 50 + 21 = 71 \]

3. Finally, subtract \( 71 \) from \( 83 \):
\[ 83 - 71 = 12 \]

### Final Result
Thus, the value of \( a^2 + b^2 + c^2 - ab - bc - ca \) is:

\[ \boxed{12} \]