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

Step 1: Identify the worst-case scenario.
There are only two colors of socks in the drawer: black and brown. When picking socks blindly, we must think about the worst possible sequence of picks, since we need to be certain, not just likely, of getting a matching pair.
Step 2: Apply the pigeonhole principle.
Imagine the unluckiest possible draw. The first sock picked can be either color, say black. The second sock picked, in the worst case, is the opposite color, brown. Now we have one black and one brown sock, no pair yet.
Step 3: Consider the third sock.
Since there are only two colors available, the third sock picked must be either black or brown. Whichever color it is, it will match one of the first two socks already picked. So after 3 picks, a matching pair is guaranteed.
Step 4: Confirm this is the minimum.
With only 2 socks picked, it is entirely possible, in the worst case, to have one black and one brown, meaning no pair yet. So 2 socks do not guarantee a pair. Therefore 3 is the smallest number that guarantees at least one matching pair, regardless of the actual counts of 10 black and 10 brown socks, since only 2 colors exist.
Step 5: Match with the options.
This corresponds to option (4), 3 pieces only. Options (1), (2), and (3) all suggest far more socks than necessary; the pigeonhole principle guarantees a pair much sooner.