A man owns socks of two colours: black and brown. Altogether, he has 30 black socks and 10 brown socks mixed in a drawer. He has to pick socks from the drawer in the dark, without seeing their colour. What is the minimum number of socks he must take out to be absolutely sure that he has at least one matching pair of the same colour?

Introduction / Context:
This is a classic Pigeonhole Principle problem involving socks of two different colours. You must determine how many socks must be drawn blindly to guarantee at least one matching pair of the same colour, regardless of the worst-case arrangement in the drawer.

Given Data / Assumptions:

• There are only two colours of socks: black and brown.

• The drawer contains 30 black socks and 10 brown socks mixed together.

• The man picks socks in the dark, so each choice is effectively random with respect to colour.

• We are asked for the minimum number of socks that must be picked to be sure of having at least one pair of the same colour.

• We assume that “pair” means two socks of identical colour, not necessarily originally sold as a pair.

Concept / Approach:
The Pigeonhole Principle states that if you distribute more objects (pigeons) than containers (pigeonholes), at least one container must contain more than one object. Here, the “pigeonholes” are the colours (black and brown), and the “pigeons” are the socks drawn. We want to find the smallest number of draws that forces at least one colour to appear at least twice.

Step-by-Step Solution:
Step 1: There are 2 colours: black and brown. Step 2: Consider the worst-case scenario. The man wants to avoid having a matching pair for as long as possible. That would mean he picks one black sock and one brown sock first. Step 3: After drawing 2 socks, the worst case is that he has 1 black and 1 brown. He has no pair yet. Step 4: The next (third) sock he picks must be either black or brown, because only these two colours exist. Step 5: Either it matches the black sock (giving two black socks) or it matches the brown sock (giving two brown socks). In either case, he now has a matching pair. Step 6: Therefore, drawing 3 socks guarantees at least one pair of the same colour.

Verification / Alternative check:
We can reason in another way. To avoid a pair, he can have at most 1 sock of each colour: 1 black and 1 brown. That is 2 socks. As soon as he draws one more sock, he must have at least two socks of one colour, because there are only two colour “categories”. Therefore, the minimum number needed to ensure a pair is 2 + 1 = 3.

Why Other Options Are Wrong:
Option D (2 socks) is wrong because he could pick one black and one brown, which do not form a matching pair.
Options B (4 socks) and C (5 socks) are larger than necessary; by the time he has picked 3 socks, he is already guaranteed to have a pair, so requiring 4 or 5 is not minimal.

Common Pitfalls:
Some examinees look at the large numbers 30 and 10 and think they must be used directly, leading them to overcomplicate the problem. However, the actual counts do not matter as long as there are at least two socks of each colour. Others forget that we are looking for a guarantee in the worst-case scenario and instead assume some average behaviour.

Final Answer:
Thus, the man must draw at least 3 socks to be absolutely sure of having a matching pair of the same colour.