In the expression 5 * 4 * 3 * 2 * 1, * is chosen from +, –, × each at most two times. What is the smallest non-negative value of the expression?

Correct Option

The problem requires finding the smallest non-negative value of the expression 5 * 4 * 3 * 2 * 1, where the asterisk (*) can be replaced by the operators +, –, or ×. Each operator can be used at most two times.

To achieve the smallest non-negative value, we aim for zero or the smallest possible positive integer.

Consider the following arrangement of operators:

5 – 4 – 3 + 2 × 1

Applying the order of operations (multiplication first, then addition/subtraction from left to right):

• First, perform the multiplication: 2 × 1 = 2
• The expression becomes: 5 – 4 – 3 + 2
• Next, perform subtractions from left to right:
• 5 – 4 = 1
• 1 – 3 = -2
• Finally, perform the addition: -2 + 2 = 0

This arrangement yields a value of 0. The operators used are two '–', one '+', and one '×', which adheres to the condition that each operator is used at most two times. Since 0 is the smallest non-negative integer, and it is achievable, it is the correct answer.

Incorrect Options

Options 1 (3), 2 (2), and 3 (1) represent positive integer values. Since it is possible to achieve a value of 0, which is smaller than any positive integer, these options cannot be the smallest non-negative value. Therefore, they are incorrect.