Permutations of the Word “Harshita” Starting with a Vowel and Ending with a Consonant
The word "Harshita" consists of 9 unique letters, with 2 A's, 3 H's, 1 I, 1 R, 1 S, and 1 T. In this detailed SEO-optimized guide, we will explore the number of permutations where these letters form words beginning with a vowel and ending with a consonant. Understanding the mathematical principles behind this problem and mastering the techniques for solving such word permutation puzzles will be valuable for students and professionals in the field of combinatorics and mathematical analysis.
Understanding the Problem
The primary objective is to find the permutations of all letters in "Harshita" that start with a vowel and end in a consonant. The vowels in "Harshita" are A and I, while the consonants are H, R, S, and T. The total number of permutations can be calculated by considering the different cases based on the starting vowel and the ending consonant. Below, we will break down the problem step by step, ensuring each permutation is thoroughly explained and matched with the appropriate mathematical formula.
Case 1: Beginning with A
The first digit can be A, and we need to find the permutations of the remaining 8 letters, excluding one A, and starting from the remaining letters: AAHHIRST. Since there are two A's, the formula for permutations in this case is given by 8! / 2!.
Calculation: 8! / 2! 40320 / 2 20160
Case 2: Beginning with I
The first digit can be I, and we need to find the permutations of the remaining 8 letters, starting from the remaining letters: AAHHRSTH. Here, there are two A's and two H's. The formula for permutations is given by 8! / (2! * 2!).
Calculation: 8! / (2! * 2!) 40320 / (2 * 2) 10080
Case 3: Beginning with H
The first digit can be H, and we need to find the permutations of the remaining 8 letters, excluding one H, and starting from the remaining letters: AARSHITT. Since there are three H's, the formula for permutations is given by 8! / 3!.
Calculation: 8! / 3! 40320 / 6 6720
Case 4: Beginning with R
The first digit can be R, and we need to find the permutations of the remaining 8 letters, and starting from the remaining letters: AAASHHITS. Here, there are three H's. The formula for permutations is given by 8! / 3!.
Calculation: 8! / 3! 40320 / 6 6720
Case 5: Beginning with S
The first digit can be S, and we need to find the permutations of the remaining 8 letters, and starting from the remaining letters: AAARHHTI. Here, there are three H's. The formula for permutations is given by 8! / 3!.
Calculation: 8! / 3! 40320 / 6 6720
Case 6: Beginning with T
The first digit can be T, and we need to find the permutations of the remaining 8 letters, and starting from the remaining letters: AAARHHIS. Here, there are three H's. The formula for permutations is given by 8! / 3!.
Calculation: 8! / 3! 40320 / 6 6720
Combining All Cases
For each starting vowel, the total number of permutations is the sum of the permutations for each starting digit. Adding up all the cases, we get:
2 * 20160 2 * 10080 4 * 6720 40320 20160 26880 87360
In conclusion, the total permutations of the letters of the word "Harshita" that begin with a vowel and end with a consonant are 87,360. This solution provides a comprehensive approach to solving word permutation problems, which can be valuable for those preparing for competitive exams or researching combinatorics in their studies.
Key Takeaways and SEO Strategy
1. **Permutations**: Understanding the concept of permutations is essential for solving problems involving the arrangement of objects. This includes knowing the formula for permutations when objects are not distinct and when they are repeated.2. **Combinatorics**: The application of combinatorial methods to solve word problems is a fundamental skill in mathematics, which can be useful in various fields including computer science, statistics, and data analysis.3. **SEO Optimization**: For this specific SEO-optimized article, the keywords like "permutations," "word problems," and "combinatorics" should be strategically used throughout the content, including meta descriptions, headings, and body text. This approach helps improve the article's visibility on search engines like Google and ensures that it ranks well in relevant search queries.
By following these SEO strategies and mastering the mathematical concepts described, you can enhance both your problem-solving skills and your online presence. This article aims to provide a thorough understanding and practical application of combinatorial principles, making it a valuable resource for students and professionals in the field.