Permutations of the Word 'DOLPHIN' with Specific Letter Constraints

The English language offers a rich field for exploring mathematical concepts through its words. Today, we will delve into the permutations of the word DOLPHIN, focusing on a specific constraint—each arrangement must start and end with a consonant. This problem not only demonstrates the principles of permutations but also provides an opportunity to explore the intricacies of combinatorial mathematics.

Let's break down the process of finding the number of such arrangements:

Step 1: Identifying Consonants and Vowels

The word DOLPHIN consists of 7 letters, 5 of which are consonants (D, L, P, H, N) and 2 are vowels (O, I).

Step 2: Choosing the First and Last Consonants

The first step in determining the number of possible arrangements is to choose a consonant for the first and a consonant for the last position. Since there are 5 consonants in total, we have 5 choices for the first position. After selecting the first consonant, we have 4 remaining consonants to choose from for the last position.

The number of ways to choose the first and last consonants is:

5 (for the first) times; 4 (for the last) 20 ways

Step 3: Arranging the Remaining Letters

After placing consonants at the start and end, we have 5 remaining letters consisting of 1 consonant and 2 vowels. These 5 letters need to be arranged in the middle 5 positions.

The number of ways to arrange 5 letters is given by the factorial of 5, denoted as 5!:

5! 5 times; 4 times; 3 times; 2 times; 1 120

Step 4: Calculating the Total Number of Arrangements

The total number of arrangements is the product of the number of ways to choose the first and last consonants and the number of ways to arrange the remaining 5 letters:

Total arrangements 20 times; 120 2400

Conclusion

Thus, the total number of permutations of the word DOLPHIN where the arrangement starts and ends with a consonant is 2400.

Further Exploration

This problem highlights the application of combinatorial mathematics and demonstrates how principles of permutations can be used to solve real-world problems. By breaking down the problem into steps and understanding each component, we can effectively find the solution.

Understanding permutations is crucial in fields such as data science, cryptography, and even in understanding natural language processing and artificial intelligence, where the ordering of elements can significantly impact outcomes.

Key Takeaways

Each permutation of the word DOLPHIN where the first and last letters are consonants results in 2400 different arrangements. The total number of arrangements is calculated by multiplying the number of ways to choose the first and last consonants by the number of ways to arrange the remaining letters. This problem showcases the application of combinatorial principles in solving real-world problems.