Competition problem — Combinatorics

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Combinatorics Difficulty 4.9 AIME Prove it United States

Problem:

You are in a completely dark room with a drawer containing 10 red, 20 blue, 30 green, and 40 khaki socks. What is the smallest number of socks you must randomly pull out in order to be sure of having at least one of each color?

Solution

Solution:

91. The maximum number of socks that can be pulled out without representing every color is $20$ blue $+30$ green $+40$ khaki $=90$ , so $91$ is the minimum needed to ensure that this doesn't happen.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.