If `a = 299, b = 298 , c = 297 ` then the value of `2a ^(3) + 2b ^(3) + 2c ^(3) - 6 abc ` is

Options

A. 5154
B. 5267
C. 5364
D. 5456

Answer

To solve the expression \(2a^3 + 2b^3 + 2c^3 - 6abc\) given \(a = 299\), \(b = 298\), and \(c = 297\), we can follow these steps:

### Step 1: Substitute the values of \(a\), \(b\), and \(c\)
We start by substituting the values into the expression:
\[ 2(299)^3 + 2(298)^3 + 2(297)^3 - 6(299)(298)(297) \]

### Step 2: Calculate \(a + b + c\)
First, we calculate:
\[ a + b + c = 299 + 298 + 297 = 894 \]

### Step 3: Calculate \(a - b\), \(b - c\), and \(c - a\)
Next, we calculate the differences:
\[ a - b = 299 - 298 = 1 \]
\[ b - c = 298 - 297 = 1 \]
\[ c - a = 297 - 299 = -2 \]

### Step 4: Calculate the squares of the differences
Now, we calculate the squares:
\[ (a - b)^2 = 1^2 = 1 \]
\[ (b - c)^2 = 1^2 = 1 \]
\[ (c - a)^2 = (-2)^2 = 4 \]

### Step 5: Substitute into the formula for \(a^3 + b^3 + c^3 - 3abc\)
We can use the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c) \left((a - b)^2 + (b - c)^2 + (c - a)^2\right) \]
Substituting the values we found:
\[ a^3 + b^3 + c^3 - 3abc = 894 \left(1 + 1 + 4\right) = 894 \times 6 = 5364 \]

### Step 6: Calculate \(2(a^3 + b^3 + c^3 - 3abc)\)
Now, we multiply by 2:
\[ 2(a^3 + b^3 + c^3 - 3abc) = 2 \times 5364 = 10728 \]

### Step 7: Calculate \(6abc\)
Next, we calculate \(6abc\):
\[ abc = 299 \times 298 \times 297 \]
Calculating \(abc\):
\[ 299 \times 298 = 89002 \]
\[ 89002 \times 297 = 26402994 \]
Now, calculate \(6abc\):
\[ 6abc = 6 \times 26402994 = 158417964 \]

### Step 8: Final Calculation
Finally, we substitute back into the expression:
\[ 2a^3 + 2b^3 + 2c^3 - 6abc = 10728 - 158417964 \]
Calculating this gives:
\[ 10728 - 158417964 = -158407236 \]

Thus, the final value is:
\[ \boxed{-158407236} \]