Question

A palindrome is a string that reads the same forward and as it reads backward. An example (if blanks and punctuation are ignored) is: A man, \(a\) plan, \(a\) canal. Panama. How many \(n\) -letter palindromes can be formed using the alphabet \\{0,1\\}\(?\)

Short answer

The number of \(n\)-letter palindromes is \(2^{\lceil n/2 \rceil}\).

Step-by-step solution

Understand a Palindrome

A palindrome is a string that remains the same when its characters are reversed. For instance, the word "radar" is a palindrome because it reads the same forwards and backwards. Our task is to find the number of palindromes given the constraints.

Analyze the Structure of an n-Letter Palindrome

A palindrome of length \(n\) will have the first half of its string determine the second half. For an even length \(n\), the first \(n/2\) characters mirror the last \(n/2\) characters. For an odd length \(n\), the middle character is symmetrical, and the first \((n-1)/2\) characters mirror these in the latter half.

Calculate the Number of Choices

The alphabet is \{0, 1\}. Hence, for each character in the first half of the palindrome (including possibly the central character if \(n\) is odd), there are 2 choices (either '0' or '1'). The number of such characters is \(\lceil n/2 \rceil\).

Apply the General Formula

Knowing each position's possible choices, the number of distinct \(n\)-letter palindromes can be calculated as the power of the number of choices for one half. Thus, the solution is \(2^{\lceil n/2 \rceil}\), reflecting all possible combinations of characters in the first half of the palindrome, which directly determines the second half.

Key concepts

Discrete Mathematics

Discrete mathematics involves mathematical structures that are fundamentally separate or distinct. Unlike continuous mathematics that deals with real numbers and smooth curves, discrete mathematics focuses on objects like integers, graphs, and statements in logic. It's particularly useful in computer science since computers operate in binary (distinct 0s and 1s). Understanding discrete mathematics is crucial for tackling problems that involve distinct or separate values, such as counting binary palindromes.

• It deals with countable, separated quantities.
• Common topics include logic, set theory, and combinatorics.
• Used extensively in algorithm design and cryptography.

Discrete mathematics lays the foundation for analyzing structures that can clearly be defined in separate states, much like the binary strings discussed in this exercise.

Binary Strings

Binary strings are sequences composed solely of the digits 0 and 1. They are fundamental to computer systems for encoding data and programming instructions. When discussing binary strings, several important characteristics are considered:

• Length: The number of individual bits (0s or 1s), which in our problem is represented by the variable \( n \).
• Structure: For binary palindromes, the arrangement of 0s and 1s must mirror themselves around the center.
• Applications: Used in digital circuitry, data transmission, and error detection.

In this exercise, binary strings are used to form palindromes. When the string reads the same forwards and backwards, it simplifies the counting process, as understanding a mere half of the string is sufficient to determine the rest.

Combinatorics

Combinatorics is the mathematical study of counting, arranging, and combining objects. It is fundamental in determining the number of ways specific arrangements can occur. For instance, in our exercise about binary palindromes, combinatorics helps us find the number of distinct palindromes possible.

• Combinatorial analysis involves calculating the number of possible outcomes.
• Basic principles include the Rule of Sum and the Rule of Product.
• Used in solving problems related to puzzles, games, and even biological structures.

In the case of palindromes, combinatorics simplifies the task to determining possible combinations for half of the string, effectively reducing complexity and making it easier to find the solution. By raising the number of character options \( (2) \) to the power of half the palindrome's length \( (\lceil n/2 \rceil) \), we leverage combinatorial principles to count possibilities efficiently.