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Interactive Audio Lesson
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Introduction to Pigeonhole Principle
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Today, we’re going to explore the pigeonhole principle and how it can help us find disjoint groups among a set of people.
What exactly is the pigeonhole principle?
It states that if you have more items than containers, at least one container must hold more than one item. In our case, the items are the age sums, and the containers are the groups we create from the individuals.
So how does this relate to finding groups of people?
Great question! If we have 9 people, we can form 511 non-empty groups. If the possible age sums are limited, it guarantees there will be some groups with the same sum.
Could you give an example?
Sure! If we have selected individuals with distinct ages that sum to various totals, some groups will inevitably share age sums, given the limited range of possible sums.
How do we ensure those groups are disjoint?
If two groups contain the same person, we can remove the shared individual and still maintain the equality of their sums, creating disjoint groups.
To sum up, the pigeonhole principle helps us guarantee the existence of disjoint groups with equal age sums.
Calculating Age Sums
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Now let’s establish the possible sums of ages among our 9 individuals.
What is the minimum sum we can have?
The minimum sum occurs when we only have one person aged 18. So that minimum is 18.
And what about the maximum sum?
The maximum is when all 9 are aged 58, giving us a total sum of 522.
So, what's the range of possible sums then?
The range is from 18 to 522. Therefore, the total range of possible sums is 505.
How does that play into our earlier discussion?
We have 511 ways to select groups from our 9 individuals, yet only 505 possible distinct sums. Thus, the pigeonhole principle guarantees at least some groups will share a sum, which leads us back to ensuring they can be disjoint.
We now understand the calculation behind our age sums before moving to the next proof step.
Ensuring Disjoint Groups
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To conclude our topic, let’s review how removing shared members helps in creating disjoint groups.
What if the two groups are already disjoint?
If they are disjoint, there's no need for modification, and we have fulfilled our requirement.
And if they’re not disjoint?
In that case, we simply remove the common members, and the sums of both groups will still remain equal.
Can you summarize why this approach works?
Absolutely! By having more groups than sums, we are guaranteed overlaps, and by removing overlaps, we ensure that the two groups become disjoint yet maintain equal sums.
This really shows the power of the pigeonhole principle!
Indeed, it showcases the cleverness of combinatorial arguments in proofs. Always remember, when items exceed containers, overlaps are inevitable!
Introduction & Overview
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Quick Overview
Standard
This section provides a mathematical proof utilizing the pigeonhole principle, demonstrating that among a selection of 9 people with ages ranging from 18 to 58, it is possible to find two disjoint groups whose ages sum to the same total. The discussion involves detailed reasoning and the application of set theory principles.
Detailed
Detailed Summary
In this section, we delve into a fascinating application of the pigeonhole principle to demonstrate that, given 9 distinct ages drawn from a range of 18 to 58, we can always find two distinct groups of individuals whose combined age sums are equal. The proof begins with an understanding of the total number of non-empty subsets we can form from these 9 individuals, which amounts to 511.
Key Points:
- Understanding Age Sums: The minimum age sum of a group is 18, while the maximum possible sum is 522 (if all individuals are 58 years old). Therefore, the possible range of age sums is 505.
- Pigeonhole Principle Application: We discern that since the number of subsets (511) exceeds the range of possible sums (505), by the pigeonhole principle, at least two distinct groups must share an identical sum.
- Ensuring Disjoint Groups: If two selected groups share members, we can remove the common individuals from both sets while preserving the equality of their age sums, thus ensuring that the resultant groups are disjoint.
This theorem illustrates the power of combinatorial reasoning in problem-solving.
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Audio Book
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Introduction to the Problem
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In this question we want to show that irrespective of what exactly are their ages, as long as they are in the range 18 to 58, it is always possible to choose 2 disjoint groups of people out of these 9 people whose sum of ages is the same.
Detailed Explanation
The objective of this section is to prove that when you have 9 individuals aged between 18 and 58, you can always find 2 different groups among them that have the same total age. It's important to note that the two groups must be disjoint, meaning they cannot have any individuals in common.
Examples & Analogies
Imagine 9 friends at a party, each having different ages between 18 and 58. If you want to split them into two teams for a game, the task is to ensure that the total age of each team is the same, even if the teams consist of different individuals.
Applying the Pigeonhole Principle
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We will do this by pigeonhole principle. So the first thing is since we want to argue about a non-empty set of people because when I want to consider the age of the people there have to be people in the group. So I have to focus on non-empty subsets. So if I have 9 people then the number of non-empty groups that I can form out of those 9 people need not be disjoint is 511.
Detailed Explanation
To understand how many combinations of groups can be made from these 9 people, we use the concept of subsets. Each of the 9 people can either be included or not included in a subset, leading to a total of 2^9 or 512 possible subsets. However, since the empty set (where no one is selected) does not count, we subtract this, leading to 511 non-empty subsets.
Examples & Analogies
Consider each person as a light switch that can be either on or off. Each combination of switches being on represents a different team composition. Since there are 9 switches, the total combinations of groups (or states) is 2^9. If everyone is 'on', you have a full group, and if all are 'off', you have the empty group, which you ignore for the count.
Minimum and Maximum Sum of Ages
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If I consider the minimum sum of ages possible in a group it could be 18. This is possible only when I have a group of just one person and that person has age 18. Whereas the maximum possible sum can occur when in my group I have all the 9 people and each of them has age 58. That means the range of possible sums here is 505.
Detailed Explanation
When forming groups, the smallest sum occurs when the youngest person, aged 18, is included alone, summing to 18. The largest sum happens when all 9 are included at the maximum age of 58, resulting in 9 * 58 = 522, and we determine that the maximum range must account for every possible sum from 18 up to 522, giving a total of 505 possible sums.
Examples & Analogies
Think of creating different age groups for a team, where the youngest player might bring only a score of 18 if they play alone, while having your oldest players together would yield the highest score possible. The range of potential total scores thus reflects all combinations of player ages.
Conclusion Using the Pigeonhole Principle
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So by pigeonhole principle, I can say that there always exists a pair of groups such that the sum of ages of the people in the groups is the same. But my question wants me to show that the groups should be disjoint.
Detailed Explanation
Having more subsets (511) than possible sums (505) means, according to the pigeonhole principle, at least two subsets must share the same sum of ages. However, we need to ensure these groups are disjoint. If two selected groups share members, removing the common individuals will not alter their sums, leading to a valid solution of disjoint groups.
Examples & Analogies
Imagine you have 511 different flavor combinations for ice cream (your subsets), and only 505 flavor profiles available (the sums). With so many combinations, it’s inevitable that at least two will taste the same. But if you have some friends to share, just scoop away common flavors to ensure everyone has their unique mix!
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Pigeonhole Principle: A principle used to deduce the existence of matching groups in sets where the items outnumber the containers.
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Distinct Ages: The ages of individuals that are specifically different from one another, crucial in forming groups.
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Subsets and Disjoint Sets: Understanding how to form subsets and ensuring they are disjoint is pivotal in achieving equal sums.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example of age sums: If you pick ages 18, 20, 25, 30, 32, 35, 40, 50, and 58, you can illustrate group selections forming the same sum.
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Using distinct ages like 19, 21, and 25 to create subsets demonstrating shared sums while keeping them disjoint.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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If more ages than groups there be, at least one sum you'll surely see!
📖 Fascinating Stories
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Imagine a party where 9 friends can only pick cakes that sum up to the same calories. They’ll definitely need to share some items!
🧠 Other Memory Gems
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Pigeon + Groups = Equal Sums (PGES) - Remember that shared items lead to equal sums.
🎯 Super Acronyms
DRIVE - Distinct, Remove, Increase, Verify, Equal (steps to ensure groups remain disjoint with same sums).
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Pigeonhole Principle
Definition:
A concept in combinatorics asserting that if more items are put into containers than the number of containers, at least one container must hold more than one item.
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Term: Subsequence
Definition:
A sequence derived by deleting some or no elements from another sequence without changing the order of the remaining elements.
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Term: Disjoint Sets
Definition:
Sets that have no elements in common.
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Term: Subset
Definition:
A set derived from another set by including some or all of its elements.
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Term: Distinct Elements
Definition:
Elements that are different from each other in a set.