Four-Letter Word Combinations And Palindrome Probability

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This article delves into the fascinating realm of combinatorics by examining the possibilities within four-letter word formations. We will explore the total number of such words that can be created using the English alphabet and then delve into calculating probabilities associated with specific word patterns. This exploration will not only sharpen our understanding of fundamental counting principles but also showcase how these principles apply in practical scenarios. Through this exploration, we aim to understand the vast landscape of possible word combinations and the likelihood of encountering specific arrangements.

(a) Determining the Sample Space: How Many Four-Letter Words Exist?

To tackle the first question, which asks us to determine the size of the sample space S, we need to calculate the total number of possible four-letter words that can be formed using lowercase alphabetic characters. This question lies at the heart of combinatorics, specifically the concept of permutations with repetition. We are allowed to repeat letters, which significantly expands our possibilities. Let's break down the calculation:

  • Understanding the Building Blocks: The English alphabet consists of 26 lowercase letters. These are our fundamental building blocks for creating words.
  • Positions in the Word: We are forming four-letter words, meaning we have four distinct positions to fill: first, second, third, and fourth.
  • Choices for Each Position: For the first position, we have 26 possible choices (any letter from 'a' to 'z'). Crucially, since repetition is allowed, we also have 26 choices for the second position, 26 choices for the third position, and 26 choices for the fourth position.
  • The Fundamental Counting Principle: The fundamental counting principle states that if there are n ways to do one thing and m ways to do another, then there are n * m* ways to do both. We extend this principle to our four positions. The total number of four-letter words is the product of the possibilities for each position.
  • The Calculation: Therefore, the total number of four-letter words is 26 * 26 * 26 * 26, which is 26 raised to the power of 4 (264).
  • The Result: Calculating this gives us 264 = 456,976. This is the size of our sample space, S. It signifies that there are 456,976 distinct four-letter words that can be formed using lowercase letters with repetition allowed. This vast number highlights the combinatorial explosion that occurs even with relatively short words.

This result underscores the power of the fundamental counting principle and provides a foundation for calculating probabilities in subsequent parts of the problem. Knowing the total number of possibilities allows us to determine the likelihood of specific events, such as the occurrence of palindromes or words with specific letter patterns. This initial calculation is a critical step in understanding the landscape of possible four-letter word combinations.

(b) Probability of a Palindrome: What are the Odds?

Now, let's transition to the second part of the problem, which focuses on calculating the probability of generating a palindrome. Palindromes are words (or phrases) that read the same forwards and backward, such as "madam" or "rotor." This constraint significantly reduces the number of favorable outcomes compared to the total sample space we calculated earlier. To determine this probability, we need to first figure out how many four-letter palindromes exist and then divide that by the total number of four-letter words.

  • Understanding Palindrome Structure: A four-letter palindrome has a specific structure: The first and fourth letters must be the same, and the second and third letters must be the same. For instance, in the palindrome "abba," the first 'a' and the last 'a' are identical, and the two 'b's are identical.
  • Choices for the First Letter: Since we're using lowercase letters, we have 26 choices for the first letter (any letter from 'a' to 'z'). This choice automatically determines the fourth letter, as it must be the same to maintain the palindrome structure.
  • Choices for the Second Letter: Similarly, we have 26 choices for the second letter. This choice automatically determines the third letter, for the same palindromic reason.
  • Calculating the Number of Palindromes: To find the total number of four-letter palindromes, we multiply the number of choices for the first letter by the number of choices for the second letter. This is because the choices for the third and fourth letters are already determined by the first two.
  • The Calculation: Therefore, the number of four-letter palindromes is 26 * 26, which equals 262.
  • The Result: Calculating this gives us 262 = 676. This means there are 676 distinct four-letter palindromes possible using lowercase letters.
  • Calculating the Probability: Now that we know the number of palindromes (676) and the total number of four-letter words (456,976), we can calculate the probability of generating a palindrome randomly. The probability is the number of favorable outcomes (palindromes) divided by the total number of possible outcomes (all four-letter words).
  • The Probability Calculation: Probability (Palindrome) = (Number of Palindromes) / (Total Number of Four-Letter Words) = 676 / 456,976.
  • Simplifying the Probability: This fraction can be simplified. 676 / 456,976 ≈ 0.00148.
  • The Final Answer: The probability of randomly generating a four-letter palindrome is approximately 0.00148, or about 0.148%. This is a relatively small probability, indicating that palindromes are much rarer than general four-letter words. This calculation highlights the power of probability in quantifying the likelihood of specific events within a larger sample space.

In conclusion, this problem has shown us how to calculate the size of a sample space in a combinatorial setting and how to determine the probability of a specific event (generating a palindrome) within that space. The principles used here—the fundamental counting principle and the definition of probability—are fundamental tools in mathematics and have wide applications in various fields, from computer science to cryptography.