Forming 4-Letter Words with Alternating Consonants and Vowels: A Comprehensive Guide

The task of forming 4-letter words with alternating consonants and vowels, where no repetitions are allowed, presents a unique challenge in the field of combinatorial mathematics. In this article, we will explore the process step-by-step to determine the total number of such words that can be created. Our approach will involve detailed calculations to ensure accuracy and provide insights into the underlying mathematical principles.

Understanding the Problem

The problem requires us to create 4-letter words where consonants and vowels alternate. Two possible patterns can be formed:

CVCV (Consonant-Vowel-Consonant-Vowel) VCVC (Vowel-Consonant-Vowel-Consonant)

Identifying the Alphabet Components

First, we need to define the structure of the words based on the order of consonants (C) and vowels (V) from the English alphabet. There are a total of 26 letters in the English alphabet, consisting of 5 vowels (A, E, I, O, U) and 21 consonants.

Step-by-Step Solution

Step 1: Determining the Number of Vowels and Consonants

In this scenario, we have:

5 vowels 21 consonants

Step 2: Calculating for Each Pattern

Pattern 1: CVCV

Choosing the Consonants: Select 2 consonants from 21. The number of ways to choose 2 consonants is ( binom{21}{2} ). The number of arrangements of these 2 consonants is ( 2! ). Choosing the Vowels: Select 2 vowels from 5. The number of ways to choose 2 vowels is ( binom{5}{2} ). The number of arrangements of these 2 vowels is ( 2! ).

Total for CVCV:

begin{align*} text{Total for CVCV} binom{21}{2} times 2! times binom{5}{2} times 2! 210 times 2 times 10 times 2 8400 end{align*}

Pattern 2: VCVC

Choosing the Vowels: Select 2 vowels from 5. The number of ways to choose 2 vowels is ( binom{5}{2} ). The number of arrangements of these 2 vowels is ( 2! ). Choosing the Consonants: Select 2 consonants from 21. The number of ways to choose 2 consonants is ( binom{21}{2} ). The number of arrangements of these 2 consonants is ( 2! ).

Total for VCVC:

begin{align*} text{Total for VCVC} binom{5}{2} times 2! times binom{21}{2} times 2! 10 times 2 times 210 times 2 8400 end{align*}

Step 3: Combining the Totals

The total number of 4-letter words that can be formed by alternating consonants and vowels is the sum of the totals for both patterns:

begin{align*} text{Total} text{Total for CVCV} text{Total for VCVC} 8400 8400 16800 end{align*}

Final Calculation and Conclusion

After calculating the values, we find the total number of 4-letter words that can be formed with alternating consonants and vowels without repetition is as follows:

begin{align*} text{Total} boxed{16800} end{align*}

This means there are 16,800 unique 4-letter words that can be formed under the specified constraints using the English alphabet.

Conclusion

This article has provided a detailed breakdown of the process of forming 4-letter words with alternating consonants and vowels. By understanding the underlying combinatorial principles, we can solve similar problems in a structured and methodical manner, enhancing our knowledge in combinatorial mathematics.