Pigeons

Definition

In the context of the pigeonhole principle, 'pigeons' refer to the items or objects being placed into containers or categories, known as 'pigeonholes.' This concept illustrates that if more items are distributed than there are containers, at least one container must hold more than one item. The pigeonhole principle is a fundamental idea in combinatorics and helps to demonstrate various concepts of counting and distribution.

5 Must Know Facts For Your Next Test

1. The pigeonhole principle can be applied in various scenarios, such as proving that in any group of people, at least two must share the same birthday if there are more than 365 people.
2. This principle is often used in computer science for problems related to data storage and distribution.
3. In its simplest form, if you have 10 pigeons and only 9 pigeonholes, at least one pigeonhole must contain at least 2 pigeons.
4. The pigeonhole principle is not just limited to physical objects; it can also apply to abstract concepts like numbers and functions.
5. The principle is a foundational idea in proofs by contradiction, demonstrating how assumptions can lead to inevitable conclusions.

Review Questions

• How does the pigeonhole principle illustrate the concept of distribution among limited resources?
  • The pigeonhole principle shows that when distributing more items than available containers, some containers will inevitably contain more than one item. This reflects real-world scenarios where resources are finite, and demand exceeds supply. For instance, if a classroom has 30 students but only 29 desks, at least one desk must be shared by two students, highlighting issues of resource allocation.
• In what ways can the pigeonhole principle be applied in combinatorial problems?
  • The pigeonhole principle is a powerful tool in combinatorics that can solve problems related to counting and arrangements. For example, it can demonstrate that among any group of people larger than the number of available birthdays, at least two individuals will share a birthday. It provides a logical framework for establishing the existence of certain conditions without needing to enumerate every possibility.
• Evaluate a situation where the pigeonhole principle leads to a counterintuitive result, explaining why this outcome occurs.
  • Consider a scenario with 13 pairs of socks placed randomly in 12 drawers. The pigeonhole principle indicates that at least one drawer will contain at least two pairs of socks. This counterintuitive result arises because although there are 12 drawers (pigeonholes), the total number of socks exceeds this limit with 13 pairs. The unexpected outcome reinforces how distributing more items than available spaces can lead to overlaps, emphasizing the necessity of adequate resources for each individual item.

Related terms

Pigeonhole Principle:

A mathematical principle stating that if n items are put into m containers and if n > m, at least one container must hold more than one item.

Combinatorics: A branch of mathematics dealing with combinations, arrangements, and counting of objects.

Cardinality: A concept that describes the size or number of elements in a set, which can be finite or infinite.