If a, b, c are distinct positive integers such that `ab+bc+cage74`, then the minimum value of `a^(3)+b^(3)+c^(3)-3abc,` is

Options

A. 42
B. 46
C. 45
D. 48

Explanation

To solve the problem, we need to find the minimum value of the expression \( a^3 + b^3 + c^3 - 3abc \) given that \( ab + bc + ca \geq 74 \) and \( a, b, c \) are distinct positive integers.

### Step 1: Understand the Expression
The expression \( a^3 + b^3 + c^3 - 3abc \) can be rewritten using the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
This means we need to minimize \( (a+b+c)(a^2 + b^2 + c^2 - ab - ac - bc) \).

### Step 2: Choose Distinct Positive Integers
Since \( a, b, c \) must be distinct positive integers, we can start by choosing small values for \( a, b, c \). Let's try \( a = 4, b = 5, c = 6 \).

### Step 3: Check the Condition
Now we need to check if these values satisfy the condition \( ab + bc + ca \geq 74 \):
\[ ab = 4 \times 5 = 20 \]
\[ bc = 5 \times 6 = 30 \]
\[ ca = 6 \times 4 = 24 \]
Now, sum these values:
\[ ab + bc + ca = 20 + 30 + 24 = 74 \]
This satisfies the condition \( ab + bc + ca \geq 74 \).

### Step 4: Calculate the Expression
Now we calculate \( a^3 + b^3 + c^3 - 3abc \):
\[ a^3 = 4^3 = 64 \]
\[ b^3 = 5^3 = 125 \]
\[ c^3 = 6^3 = 216 \]
Now sum these:
\[ a^3 + b^3 + c^3 = 64 + 125 + 216 = 405 \]
Next, calculate \( 3abc \):
\[ abc = 4 \times 5 \times 6 = 120 \]
So,
\[ 3abc = 3 \times 120 = 360 \]
Now, substitute these values into the expression:
\[ a^3 + b^3 + c^3 - 3abc = 405 - 360 = 45 \]

### Conclusion
Thus, the minimum value of \( a^3 + b^3 + c^3 - 3abc \) given the condition is \( \boxed{45} \).