1. **State the problem:** James wants to remove balls from a bag without looking, to guarantee he has at least 4 balls of different colors (red, blue, black, yellow). 2. **Given:** - Red balls = 10 - Blue balls = 13 - Black balls = 20 - Yellow balls = 25 3. **Goal:** Find the minimum number of balls James must remove to be sure he has at least one ball of each of the 4 colors. 4. **Approach:** The worst-case scenario is removing as many balls as possible without getting all 4 colors. 5. **Worst case:** He could remove all balls of 3 colors only and none of the 4th color. 6. To avoid getting 4 different colors, he might remove all balls of the three colors with the largest counts plus some balls from the smallest color to prevent hitting 4 different colors. 7. The counts are: Red (10), Blue (13), Black (20), Yellow (25). 8. To maximize removal without 4 colors, remove all from 3 colors: - Black = 20 - Yellow = 25 - Blue = 13 Total = 20 + 25 + 13 = 58 balls 9. At this point, he hasn't removed any red balls, so only 3 colors are present. 10. On removing one more ball (ball number 59), it must be red (because others are exhausted), so he definitely gets 4 different colors. 11. Hence, the minimum number of balls to remove is $$58 + 1 = 59$$. **Final answer:** James must remove at least **59 balls** to guarantee he has 4 balls of different colors.