Problem 30.6 Easy Free

Socks for a Matching Pair

A drawer holds 2 red socks, 4 yellow socks, and 5 blue socks. Socks are drawn one at a time without looking at colour. What is the minimum number of socks you must draw to be certain of having two socks of the same colour?

Work it out, then compare with the answer.

SolutionShow the worked solution

There are three colours. With $4$ socks and only $3$ colours available, the pigeonhole principle forces two socks to share a colour.

Three socks are not enough. Each colour has at least one sock, so it is possible to draw one red, one yellow and one blue sock, which gives no matching pair.

Hence the minimum number that guarantees a matching pair is

\boxed{4}

Key idea. With $k$ colours, each present in the drawer, $k+1$ socks guarantee a pair, and $k$ do not.

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