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Chapter 3: Problem 2

Show that in any 27 -letter word, at least two letters are the same.

Short Answer

Expert verified
Using the Pigeonhole Principle, we have 27 letters (pigeons) and only 26 distinct available letters in the alphabet (pigeonholes). Since there are more pigeons than pigeonholes, and each pigeon must occupy a pigeonhole, at least one pigeonhole must contain two or more pigeons. Thus, at least one letter repeats in any 27-letter word.

Step by step solution

01

Identifying the Pigeons and Pigeonholes

In this problem, the pigeons are the 27 letters in the given word and the pigeonholes are the 26 distinct letters of the alphabet.
02

Applying the Pigeonhole Principle

Since there are 27 pigeons (letters in the word) and 26 pigeonholes (distinct letters of the alphabet), according to the Pigeonhole Principle, at least one pigeonhole must contain at least two pigeons.
03

Interpreting the result

In the context of this problem, having at least one pigeonhole containing two or more pigeons means that there is at least one letter of the alphabet that appears two or more times in the given 27-letter word. Thus, we have shown that in any 27-letter word, at least two of the letters must be the same.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discrete Mathematics
Discrete mathematics is a branch of mathematics that deals with distinct and separate values. Unlike continuous mathematics, which focuses on concepts like calculus and real numbers, discrete mathematics is concerned with objects that can be counted, ordered, and arranged. It is particularly useful in the fields of computer science, cryptography, and graph theory.

The Pigeonhole Principle is one key topic within discrete mathematics. This principle states that if you have more items than containers, then at least one container must contain more than one item. It is a simple yet powerful idea that can be applied in various scenarios, such as the exercise where we are considering 27 letters against the 26 available alphabet slots. The principle helps highlight logical and predictable outcomes from seemingly straightforward setups, proving that complexity often lies in simplicity.
Combinatorics
Combinatorics is a core area of discrete mathematics that involves the study of counting and arranging objects. It plays a vital role in how we solve problems that involve combinations, permutations, and probabilities. This field helps us to understand the finite discrete structures and models which are part of our daily computational problems.

The given exercise relies on combinatorial logic, where we recognize that arranging 27 'objects' or letters means that some must overlap when placed into only 26 available 'places' or distinct letters of the alphabet. This is a classic application of the Pigeonhole Principle, and it showcases how combinatorial reasoning leads to definite solutions by logically constraining possibilities based on given limits and conditions. Understanding these concepts prepares students for more complex problems by teaching them the skill of logical deduction.
Counting Principles
Counting principles are fundamental tools in discrete mathematics and combinatorics that offer a basis for evaluating the number of ways an event can occur. They are essential in problem-solving, providing an analytical framework for different scenarios, such as choosing items, arranging objects, or categorizing elements based on properties.

In the current exercise example, one such counting principle鈥攖he Pigeonhole Principle鈥攊llustrates why at least two letters must repeat in a 27-letter word. It emphasizes the inevitability of repetition when distributing a larger number of items into a smaller number of distinct categories. Other notable counting principles include permutations, which consider the arrangement order of objects, and combinations, which consider selections of objects without regard to the order.
  • Basic Counting Principle: Enumerate the number of all possible outcomes by multiplying the number of choices at each step.
  • Additive Principle: Combine the number of ways two or more mutually exclusive events can occur.
  • Multiplicative Principle: Multiply the number of ways multiple events can simultaneously occur.
These principles help in abstract thinking and problem-solving, skills crucial for various applications in science and engineering.

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Most popular questions from this chapter

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll}{1} & {\text { if } i=j} \\ {0} & {\text { otherwise }}\end{array}\right. $$ \(\left[\delta_{j} \text { is called Kronecker's delta, after the German mathematician Leopold }\right.\) Kronecker \((1823-1891) . ]\) $$ \sum_{i=1}^{5} \sum_{j=1}^{5} \delta_{i j} $$

Prove. A bijection exists between any two closed intervals \([a, b]\) and \([c, d],\) where \(a< b\) and \(c< d\) . (Hint: Find a suitable function that works.)

Let \(A=\left[\begin{array}{ccc}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{ccc}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right]\) . Find each. $$3 A+(-2) B$$

Prove each, where \(X \sim Y\) implies set \(X\) is equivalent to set \(Y\). If \(A \sim B,\) then \(B \sim A\) (symmetric property).

Just as \(\sum\) is used to denote sums, the product \(a_{k} a_{k+1} \ldots a_{m}\) is denoted by \(\prod_{i=k}^{m} \mathrm{a}_{i} .\) The product symbol \(\Pi\) is the Greek capital letter \(p i .\) For example, \(n !=\prod_{i=1}^{n} i .\) Evaluate each product. $$\prod_{i=1}^{3}(i+1)$$
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