Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to the Pigeon-Hole Principle
Unlock Audio Lesson
Today, we're going to discuss a fundamental concept in combinatorics known as the Pigeon-Hole Principle. Who can tell me what this principle states?
Is it about distributing items into boxes or something similar?
Yes, that's correct! If you have more items than boxes, at least one box must contain more than one item. For instance, if you have 13 pigeons and 12 holes, at least one hole will contain at least 2 pigeons.
Can you explain why that is true?
Certainly! If we try to place each pigeon into a hole and assume each hole can only contain one pigeon, we'd only be able to fit 12 pigeons into 12 holes. However, since there are 13 pigeons, we have to place at least one pigeon into an already occupied hole. This illustrates the principle well.
What if we have more than two holes?
Good question! The principle scales. If N pigeons are distributed across K holes, with N > K, then at least one hole will contain at least ⌈N/K⌉ pigeons.
Can you give us an example of that?
Of course! If you have 13 pigeons and 12 holes, each hole will contain at least ⌈13/12⌉ = 2 pigeons.
To summarize today's discussion: The Pigeon-Hole Principle ensures that if you distribute more items than containers, at least one container contains more than one item.
Proof and Generalization of the Pigeon-Hole Principle
Unlock Audio Lesson
Now, let’s delve deeper into a generalization of the Pigeon-Hole Principle. Can anyone define what the generalized principle states?
I think it says something about having an average or expected number in each hole?
Exactly! If you have N objects distributed among K boxes, at least one box must contain at least ⌈N/K⌉ objects. This is important for understanding distributions in real-world scenarios.
So for example, if there are 25 apples distributed into 6 baskets, how many apples will be in at least one basket?
Great question! Here, ⌈25/6⌉ = 5, so at least one basket must contain at least 5 apples.
How can we prove the original claim?
We can prove it via contradiction. If every hole had only one pigeon and there were 12 holes, we wouldn't be able to accommodate 13 pigeons since that would imply 13 = 12, which is a contradiction.
In summary, we've shown how the Pigeon-Hole Principle can be generalized to apply to any number of items and containers, ensuring at least one container has more than the average number of items.
Applications of the Pigeon-Hole Principle
Unlock Audio Lesson
Now that we understand the principle, let’s examine its applications. Why do you think this principle is important in real-life scenarios?
It seems like it could relate to sharing resources or organizing data.
Absolutely! It helps in determining optimal resource allocation. For instance, in networks, if there are more data requests than servers, at least one server will handle more than one request, leading to potential overload.
Are there any social implications?
Yes! In social dynamics, consider a party with 6 individuals. No matter how friendships and enmities are organized, there will always exist either three mutual friends or three mutual enemies.
That's interesting! Are there situations where this principle doesn't apply?
The principle reliably applies when conditions are fixed and there are more items than containers, but distributions can vary. Without this structure, the outcomes could be unpredictable.
To summarize, the Pigeon-Hole Principle not only applies mathematically but has broad applications across various fields, including computing and social science.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
This principle has wide-ranging applications in combinatorics and discrete mathematics. It illustrates that in situations where items are placed into containers, if the number of items exceeds the number of containers, at least one container will hold more than one item. A generalized form of this principle further extends this understanding to any number of objects and containers.
Detailed
Pigeon-Hole Principle
The Pigeon-Hole Principle states that if you have N items (pigeons) placed into K containers (pigeonholes), and if N > K, then at least one container must hold more than one item. This intuitive concept can be proved by contradiction; if each container could hold only one item, the maximum would be K items, contradicting our premise that there are N items (where N > K). The principle can be generalized to state that if N objects are distributed among K containers, then at least one container must contain at least ⌈N/K⌉ objects, where ⌈x⌉ denotes the ceiling function, which rounds x up to the nearest integer. This principle has significant implications in various fields including computer science, statistics, and information theory. An example of its application can be seen in social settings, where if there are 6 people in a party, there must be at least 3 who are either mutual friends or mutual enemies.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Basic Concept of the Pigeon-Hole Principle
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So now let us see another interesting counting rule which we encounter very often in discrete mathematics and this is called pigeon-hole principle. So what is the scenario here? So in this example you have 13 pigeons and you have 12 holes and suppose the pigeons are going to randomly occupy these 12 holes. We don't know in what order they will be going and occupying these holes. But irrespective of the way they are going to occupy these 12 holes we can say that there will always exist at least 1 hole which will have 2 or more pigeons.
Detailed Explanation
The Pigeon-Hole Principle states that if you have more items (pigeons) than containers (holes), then at least one container must hold more than one item. In the example, since there are 13 pigeons and only 12 holes, it's impossible to place each pigeon in a different hole, ensuring that at least one hole must contain at least two pigeons. This conclusion follows logically since if each of the 12 holes had just one pigeon, there would only be 12 pigeons, which leads to a direct contradiction of having 13 pigeons.
Examples & Analogies
Imagine you have 13 different colored balls and only 12 boxes to store them. When you try to place each ball in a separate box, you'll find that one box must end up containing at least two balls because there aren't enough boxes for every ball. This common scenario illustrates how the Pigeon-Hole Principle applies to everyday situations.
Generalized Pigeon-Hole Principle
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So the generalized pigeon-hole principle is the following: So imagine you have N objects, in this case you had N pigeons, and suppose those N objects are assigned to K boxes in a random fashion, then the pigeon-hole principle states that there will be at least 1 box which will have ⌈N/K⌉ many objects.
Detailed Explanation
The generalized version of the Pigeon-Hole Principle extends the basic idea to any number of objects and boxes. It states that if you distribute N objects into K boxes, then at least one box will contain at least ⌈N/K⌉ objects. The ceiling function (⌈x⌉) rounds up to the nearest integer, meaning if N is not perfectly divisible by K, some boxes will contain more items than others, demonstrating that at least one will exceed this average.
Examples & Analogies
Consider you have 20 apples to distribute among 6 baskets. According to the generalized Pigeon-Hole Principle, at least one basket will contain at least ⌈20/6⌉ = ⌈3.33⌉ = 4 apples. This leads to the effective conclusion that no matter how you distribute the apples, at least one basket must hold 4 or more.
Real-World Application of Pigeon-Hole Principle
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So let's see an application of pigeon-hole principle. […] Then our claim is that irrespective of the way the people are mutually friends or enemies there always exists either 3 mutual friends in the party or 3 mutual enemies.
Detailed Explanation
A scenario illustrates that if you have 6 people at a party, each pair can either be friends or enemies. The Pigeon-Hole Principle helps us prove that among these 6 people, there will always be either 3 who are friends with each other or 3 who are enemies. By selecting one person and analyzing their relationships with the remaining 5, we can categorize them into friends and enemies, leading us inevitably to conclude that one of these relationships must dominate (three mutual friends or three mutual enemies) due to the sheer number of individual relationships involved.
Examples & Analogies
Imagine a small gathering where you know that everyone either knows each other well (friends) or doesn’t (enemies). If there are 6 attendees, despite how those relationships unfold, you will always find either a trio that knows each other or a trio that doesn't. It's reminiscent of group dynamics in classrooms where any mix of students will form either friendships or cliques that can be analyzed using the Pigeon-Hole Principle.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Pigeon-Hole Principle: If you have more items than containers, at least one container must hold more than one item.
-
Ceiling Function: The integer value greater than or equal to a number.
-
Generalized Version: States that if N objects are assigned to K boxes, at least one box must contain at least ⌈N/K⌉ objects.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
If there are 11 pairs of socks but only 10 drawers, at least one drawer will contain at least 2 pairs of socks.
-
At a party with 8 participants, there must be at least 3 who are mutual friends or mutual enemies.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
When pigeons fly and holes are few, in a crowded hole, more than one will do.
📖 Fascinating Stories
-
In a forest, 13 pigeons found only 12 cozy holes. Each pigeon landed from high above, and to their surprise, one hole had to host a second dove, proving the Pigeon-Hole Principle.
🧠 Other Memory Gems
-
Pigeons Must Multiply In Holes (PMMIH): If more pigeons than holes, one must double up!
🎯 Super Acronyms
POC
- Pigeons Over Containers
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: PigeonHole Principle
Definition:
A principle stating that if N items are placed into K containers, with N > K, then at least one container must contain more than one item.
-
Term: Ceiling Function
Definition:
A mathematical function that rounds a number up to the nearest integer.
-
Term: Generalized PigeonHole Principle
Definition:
An extension of the Pigeon-Hole Principle stating that if N objects are distributed among K boxes, then at least one box contains at least ⌈N/K⌉ objects.
-
Term: Contradiction Proof
Definition:
A method of proving a statement by assuming the opposite is true and demonstrating that this assumption leads to a contradiction.