A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
- A)
- B)
- C)
- D)
- E)
Answer
C
Key insight
Four colors means four socks can all be different, but a fifth sock must repeat one of the four colors.
Solution
Think about the worst case. With four colors available, it is possible to draw four socks that are all different: one red, one green, one blue, one white. So four socks do not guarantee a pair.
Once a fifth sock is drawn, there are only four colors for five socks, so two of them must share a color. This is the pigeonhole principle with the colors as the "holes."
Hence socks are needed, and the answer is .
Why this works
"Guarantee" questions are about the worst case, not the likely case. First build the largest possible draw that fails (one of each color), then add one more; that "largest failure plus one" is the standard pigeonhole answer, here (number of colors) .
The trap
Answering 4 (the number of colors) instead of one more than the number of colors.
Common mistakes
- Answering 4 (the number of colors) instead of one more than the number of colors.
- Being distracted by "at least 2 of each color," which only guarantees a pair exists; it does not change the count.
Techniques
Consider the largest/smallest element or boundary case · Pigeonhole principle