Question

A drawer contains 10 black and 10 brown socks, all mixed up. What is the fewest number of socks you must take out from the drawer without looking, to be sure of getting a pair of socks of the same color?

• A. 7 pairs
• B. 7 pieces only
• C. 10 pieces only
• D. 3 pieces only

Hint:

Only 2 colors exist, think pigeonhole: worst case before a pair forms.

Answer 1

The Correct Option is D

This is a classic application of the pigeonhole principle, which can be solved with a simple formula instead of a worst-case story.

1. The pigeonhole principle states that if you distribute more than n items into n categories, at least one category must contain more than one item.
2. Here, the categories are the 2 sock colors: black and brown. We want to guarantee that at least one color has 2 socks, a matching pair, among the socks picked.
3. Using the formula for guaranteeing k items in the same category, given m categories: minimum items needed = m(k - 1) + 1.
4. Substitute m = 2 (colors) and k = 2 (we want a pair, i.e., 2 of the same color): minimum = 2(2 - 1) + 1 = 2(1) + 1 = 3.
5. So picking just 3 socks guarantees, without fail, that at least 2 of them share the same color, since it is impossible to distribute 3 socks across only 2 colors without repeating a color.
6. The actual quantities of 10 black and 10 brown socks are irrelevant to this minimum; they only matter if the question asked for a pair of a specific color.

The fewest number of socks needed is 3, matching option (4). \[\boxed{3 \text{ socks}}\]