Murari has 9 pairs of dark-blue socks and 9 pairs of black socks (all loose in one bag). If he picks three socks at random, what is the probability that he gets at least one matching pair?

Introduction / Context:With only two colours available (dark blue and black), any selection of three socks must contain at least two of the same colour by the pigeonhole principle. Therefore, a matching colour pair is guaranteed.

Given Data / Assumptions:

• Two colours: dark blue and black.
• Three socks drawn without replacement.
• Pairs do not matter; colour matches are sufficient for a “pair.”

Concept / Approach:Pigeonhole Principle: placing 3 items into 2 categories ensures at least one category receives ≥2 items.

Step-by-Step Reasoning:Possible colour-count splits for 3 socks with 2 colours are 3–0 or 2–1; both contain a colour appearing at least twice.Hence the event “at least one matching pair by colour” occurs with certainty.

Verification / Alternative check:Trying to avoid a pair would require 3 socks all of different colours—impossible with only two colours available.

Why Other Options Are Wrong:Combinatorial expressions given in other options do not evaluate to 1 and misrepresent the structure of the sample space.

Common Pitfalls:Confusing identical socks within a pair with “colour” matching; here, only colour match is needed to constitute a pair.

Final Answer:1