Question

A man has 10 black socks and 11 blue socks scrambled in a drawer. Still half- asleep. the man reaches in the drawer to get a pair of matching socks. How many socks should he select, one at a time, before he will be sure that he has a matching pair. How many selections are needed to be sure he has a blue pair?

Short answer

3 socks for a matching pair; 12 socks for a blue pair.

Step-by-step solution

Understand the Problem

We have two types of socks: black and blue. There are 10 black socks and 11 blue socks. The task is to determine the minimum number of socks he needs to pick at random to ensure he has at least one matching pair of socks and then specifically a pair of blue socks.

Calculate for a Matching Pair

If he picks one sock, it could be either black or blue with no pair. If he picks a second sock, he might get a pair if both are of the same color. But to guarantee a matching pair, picking a third sock ensures he has at least two of the same color, because of the pigeonhole principle (with two colors, three socks mean at least one pair).

Calculate for a Blue Pair

To ensure he has a blue pair specifically, consider the worst-case scenario: he picks all wrong socks first. That would mean he picks up all 10 black socks before picking blue. Therefore, he would need to pick 12 socks to guarantee getting more than one blue sock (i.e., a blue pair).

Key concepts

Combination Problems

Combination problems involve selecting items from a larger set without regard to order. In this exercise, the man needs to identify how many socks to select to ensure obtaining a matching pair, and more specifically, a pair of blue socks. Imagine he draws socks one by one. Initially, any selection is merely a matter of luck.

• Picking one sock leads to no pair, whether it's black or blue.
• Drawing a second sock offers a chance of pairing if colors match.
• To guarantee a matching pair, a third sock is necessary. This is because, per the pigeonhole principle, with two colors, three socks will ensure that at least one color repeats, forming a pair.

In summary, determining combinations involves considering possible selections and ensuring all outcomes are covered to meet the desired conditions.

Probability in Mathematics

Probability helps measure the likelihood of an event. Here, it's vital in assessing the certainty (or lack thereof) of obtaining matching socks and specifically blue ones. When you have a mixture of socks, drawing a specific pair becomes a probability problem. The steps are:

• The first sock could be any color, impacting the probability of the second forming a pair.
• With the third sock, probability guarantees a matching pair due to available choices being limited.

However, to guarantee specifically a blue pair, the worst case must be assumed:

• All black socks could be picked before selecting a blue one.
• Hence, at least 12 picks are needed to ensure getting more than one blue sock, achieving a pair.

This understanding of probability ensures all potential outcomes are anticipated for desired results.

Discrete Mathematics

Discrete mathematics involves dealing with countable, distinct elements. The problem of selecting socks is an example where discrete mathematics concepts can be applied effectively.

• The task involves a discrete set of socks - 10 black and 11 blue - which are finite and countable.
• The solution uses the Pigeonhole Principle, a key concept in discrete mathematics asserting that if there are more items than containers, at least one container must contain more than one item.
• The problem is solved by calculating boundaries: a minimum sock count ensuring conditions are met.

This form of problem-solving highlights how discrete structures, like combinations of socks, can be managed using principles and theories within this field. The methods ensure definitive measures and solutions for problems involving finite sets.