A bag contains 100 apples, 100 bananas, 100 oranges, and 100...

Name: Problem 14, Chapter 3: Introductory Combinatorics
Uploaded: 2023-09-13T05:29:00-08:00
Duration: 3 min 40 s
Channel: Aparna Shakti
Description: A bag contains 100 apples, 100 bananas, 100 oranges, and 100 pears. If I pick one piece of fruit out of the bag every minute, how tong will it be before I am assured of having picked at least a dozen pieces of fruit of the same kind?

A bag contains 100 apples, 100 bananas, 100 oranges, and 100 pears. If I pick one piece of fruit out of the bag every minute, how tong will it be before I am assured of having picked at least a dozen pieces of fruit of the same kind?

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Key Concepts

Worst-case Analysis

This analytical approach involves considering the most unfavorable distribution of picked items, one that delays meeting the desired condition for as long as possible. By assuming that the fruits are picked in such a way as to evenly distribute picks among the different types, the minimal number required to guarantee a dozen of one kind is determined.

Pigeonhole Principle

This is a fundamental concept in combinatorics which states that if you have more objects than there are containers to hold them, then at least one container must contain more than a certain number of objects. In the context of the problem, the four fruit types act as containers, and the principle guarantees that after enough fruit picks, one type will contain at least a dozen fruits.

Extremal Principle

Closely related to worst-case analysis, this principle involves identifying the extreme or boundary conditions of a scenario. In this problem, by calculating the maximum number of fruits that could be picked without having twelve of any type, the extremal condition is used to derive the minimum number required to ensure the condition is met.

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