Question

$a, b, c$ are integers, $|a| \ne |b| \ne |c|$ and $-10 \le a,b,c \le 10$. What will be the maximum possible value of $[\,abc - (a+b+c)\,]$?

• A. 524
• B. 693
• C. 731
• D. 970
• E. None of the above

Hint:

For expressions of the form $abc-(a+b+c)$ with bounded integers, aim for two large-magnitude negatives and one large positive (all with distinct absolute values) so that the product is large and the sum is negative.

Answer 1

The Correct Option is C

Step 1: Understand the goal.
We want to maximize the expression \[ abc - (a+b+c). \] To make this large, we need:

• A large positive product \(abc\).
• A small (preferably negative) sum \(a+b+c\) so that subtracting it makes the expression larger.

Strategy: Choose two negative numbers (large in magnitude) and one positive number. This gives \(abc > 0\) (positive product) and \(a+b+c < 0\) (negative sum).

Step 2: Choose numbers under constraints.
The largest allowed distinct magnitudes are \(10, 9, 8\). To satisfy \(|a|\ne|b|\ne|c|\) and keep two negatives: \[ a = -10, \quad b = -9, \quad c = 8. \] (We cannot use \(c=10\) or \(c=9\) because that would duplicate absolute values.)

Step 3: Compute the value.
\[ abc = (-10)(-9)(8) = 720 \] \[ a+b+c = -10 - 9 + 8 = -11 \] \[ abc - (a+b+c) = 720 - (-11) = 720 + 11 = \boxed{731}. \]

Step 4: Why this is maximal.

• All-positive choice \((10,9,8)\): \[ 720 - (27) = 693 < 731 \]
• Making the positive number \(9\) or \(10\) is disallowed because absolute values would repeat.
• Any other triple with smaller magnitudes or with one/three negatives either reduces \(abc\) or increases the sum, giving a value \(\leq 731\).

Final Answer:

\[ \boxed{731 \quad \text{(maximum possible value)}} \]