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?

We need to assign +, −, × to the four * positions such that each operator is used at most twice, and obtain the smallest non-negative value. Try 5 × 4 − 3 × 2 − 1 × ... wait, we have only four operator slots. Consider 5 − 4 + 3 − 2 × 1 = 5 − 4 + 3 − 2 = 2. Try 5 × 4 − 3 × 2 − 1·... only 4 operators. Try 5 + 4 − 3 × 2 − 1 = 5 + 4 − 6 − 1 = 2. Try 5 − 4 × 3 + 2 × 1 with operators (−, ×, +, ×) each used at most twice: 5 − 12 + 2 = −5 (negative, rejected). Try 5 × 4 − 3 − 2 × ... Try 5 + 4 − 3 − 2 × 1 with ops (+, −, −, ×): 5 + 4 − 3 − 2 = 4. Try 5 − 4 − 3 + 2 × 1 = 0 with ops (−, −, +, ×) each used at most twice. So 0 is achievable. Hence the smallest non-negative value is 0.