1. **Problem statement:** We have socks of 5 colors with quantities: red 90, pink 80, green 70, black 60, blue 50. We want to find the minimum number of socks to pick to guarantee at least 20 pairs (pairs = 2 socks of the same color). 2. **Understanding pairs:** A pair is 2 socks of the same color. To ensure 20 pairs, we need at least 40 socks forming these pairs. 3. **Worst-case scenario approach:** To guarantee 20 pairs, consider the worst case where we pick socks to minimize pairs. 4. **Maximum pairs per color:** - Red: 90 socks → max 45 pairs - Pink: 80 socks → max 40 pairs - Green: 70 socks → max 35 pairs - Black: 60 socks → max 30 pairs - Blue: 50 socks → max 25 pairs 5. **Worst case: pick socks to have fewer than 20 pairs:** We try to pick socks so that total pairs < 20. 6. **Max pairs if we pick $2n$ socks of a color:** Pairs = $\lfloor \frac{\text{socks}}{2} \rfloor$. 7. **To minimize pairs, pick odd number of socks per color:** Because an odd number of socks gives one less pair than the half. 8. **Try to pick socks to have 19 pairs total:** Distribute 19 pairs among colors to maximize socks picked without reaching 20 pairs. 9. **Example distribution:** - Red: 9 pairs → 19 socks (since 9 pairs = 18 socks + 1 extra sock) - Pink: 5 pairs → 11 socks - Green: 3 pairs → 7 socks - Black: 1 pair → 3 socks - Blue: 1 pair → 3 socks Total pairs = 9+5+3+1+1=19 pairs Total socks = 19+11+7+3+3=43 socks 10. **If we pick one more sock (44th sock), it must form the 20th pair:** Because all socks are accounted for in the worst case. **Final answer:** Minimum socks to pick to guarantee 20 pairs = $44$.