The Solution

1. Total Outcomes: Since each of the six dice has 6 possible results, the total number of outcomes is 6⁶ = 46,656.

2. Choosing the Numbers: We need to choose exactly 4 distinct numbers out of the 6 available. The number of ways to do this is:
C(6, 4) = 15

3. Distributing the Dice: We have 6 dice and we want to assign them to our 4 chosen numbers such that each number appears at least once. There are two ways the counts can be distributed:

• Case A: {3, 1, 1, 1} - One number appears 3 times, and the other three appear once each.
  • Choose which number appears 3 times: C(4, 1) = 4
  • Arrangements: 6! / 3! = 120
  • Total for Case A: 15 × 4 × 120 = 7,200
• Case B: {2, 2, 1, 1} - Two numbers appear twice, and the other two appear once each.
  • Choose which two numbers appear twice: C(4, 2) = 6
  • Arrangements: 6! / (2! × 2!) = 180
  • Total for Case B: 15 × 6 × 180 = 16,200

4. Final Sum: Adding the cases together gives us 7,200 + 16,200 = 23,400 favorable outcomes.

5. Probability: 23,400 / 46,656 ≈ 0.50154