Arrangements of the Word ‘Arrangement’ Starting with a Vowel: A Comprehensive Guide

Have you ever pondered the myriad ways in which the letters of the word 'arrangement' could be rearranged such that the word starts with a vowel? This article provides a detailed analysis of the various arrangements, highlighting the step-by-step process and the mathematical calculations involved. By the end of this article, you will have a thorough understanding of how to calculate the number of arrangements of a given word under specific conditions, particularly focusing on beginning with a vowel.

Introduction to the Problem

The word 'arrangement' is a fascinating example for exploring permutations, especially those that start with a specific letter. In this article, we will delve into how to determine the total number of ways to arrange the letters of the word 'arrangement' such that it begins with a vowel.

Step-by-Step Analysis

Step 1: Count the Total Letters

The word 'arrangement' consists of 11 letters: a, r, r, a, n, n, g, e, m, e, n.

Step 2: Identify the Vowels in the Word

The vowels in the word 'arrangement' are: a, a, e, e, e.

Step 3: Calculate Arrangements Starting with Each Vowel

We will start by considering each vowel in turn and calculating the number of arrangements of the remaining letters.

Case 1: Starting with 'a'

If the word starts with 'a', the remaining letters are: r, r, a, n, n, g, e, m, e, n. There are 10 letters in total with the following counts:

Letter 'r' - 2 times Letter 'a' - 1 time Letter 'n' - 2 times Letter 'g' - 1 time Letter 'e' - 2 times Letter 'm' - 1 time Letter 'n' - 2 times (counted again)

The number of arrangements can be calculated using the formula for permutations of a multiset:

Arrangements ( frac{n!}{n_1! . n_2! . . . n_k!} )

Here, ( n ) is the total number of items, and ( n_1, n_2, . . . , n_k ) are the counts of each indistinguishable item. Plugging in the values, we get:

Arrangements ( frac{10!}{2!.1!.2!.2!.1!.2!} ) ( frac{3,628,800}{8} ) 453,600

Case 2: Starting with 'e'

The second vowel, 'e', requires similar calculations:

If the first letter is 'e', the remaining letters are: a, r, r, a, n, n, g, m, e, n. There are 10 letters in total with the following counts:

Letter 'r' - 2 times Letter 'a' - 1 time (counted again) Letter 'n' - 2 times Letter 'g' - 1 time Letter 'm' - 1 time Letter 'e' - 1 time Letter 'n' - 2 times (counted again)

Using the same formula, the number of arrangements is:

Arrangements ( frac{10!}{2!.1!.2!.1!.1!.2!} ) ( frac{3,628,800}{8} ) 453,600

Total Arrangements

Summing the results of both cases, we get:

Total arrangements 453,600 453,600 907,200

Alternative Method of Calculation

Another method to approach this problem is to consider the following steps:

Choose a vowel to begin with - the choice is between 'a' and 'e', giving 2 choices. Arrange the other 10 letters (taking into account duplicates):

There are 2 'r's, 2 'n's, and 2 'e's, plus one each of 'g', 'm', and another 'a'. The number of arrangements for the 10 letters is:

Arrangements ( frac{10!}{2!.2!.2!} ) ( frac{3,628,800}{8} ) 453,600

Multiplying by the two choices for the starting vowel, we get:

Total 2 . 453,600 907,200

Conclusion

Through both methods, we have calculated the total number of ways to arrange the letters of the word 'arrangement' so that it begins with a vowel. This result highlights the power of permutations and the formula for arranging items in a multiset. Whether you prefer the step-by-step calculation of individual vowels or the alternative method of choosing a vowel and arranging the rest, the final result remains the same.

Keywords: arrangement, permutations, vowels, multiset