Problem 33 - Algebra 2

Question

How many different arrangements of the letters of the Hawaiian word aloha are possible?

Step-by-Step Solution

Answer

There are 60 different arrangements.

Step 1: Understand the Problem

The problem is asking for the number of ways to arrange the letters in the word "aloha." The word consists of five letters: A, L, O, H, and A. Out of these, the letter A appears twice, while L, O, and H each appear once.

Step 2: Use the Permutation Formula for Repeated Elements

To find the number of arrangements of letters in "aloha" where some letters are repeated, we use the permutation formula for a multiset: \[ \frac{n!}{n_1! \times n_2! \times ... \times n_k!} \] where \( n \) is the total number of letters, and each \( n_i \) is the factorial of the number of times each letter is repeated.

Step 3: Calculate the Factorial of the Total Number of Letters

The total number of letters, \( n \), in "aloha" is 5. First, calculate \( n! = 5! \): \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \].

Step 4: Calculate the Factorial for the Repeated Letter

The letter A is repeated 2 times, so we need to find \( 2! \): \[ 2! = 2 \times 1 = 2 \].

Step 5: Apply the Permutation Formula

Now apply the formula: \[ \frac{5!}{2!} = \frac{120}{2} = 60 \].

Step 6: Interpret the Result

The calculation means there are 60 different ways to arrange the letters in the Hawaiian word "aloha."

Key Concepts

Factorial

In mathematics, the concept of a factorial is crucial when dealing with permutations and combinations. The factorial of a positive integer \( n \) is denoted as \( n! \) and is the product of all positive integers less than or equal to \( n \). This is mathematically expressed as: \[ n! = n \times (n-1) \times (n-2) \times \, ... \, \times 1 \]. Each step in the multiplication involves decreasing by one until you reach one. This simple operation helps calculate how many different ways we can arrange a set of items.
For example, in the word "aloha," there are five letters, so we compute \( 5! \) as \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials grow very quickly, so they are pivotal when dealing with large numbers of items, providing the basis for other more complex equations.

Multisets

A multiset, in contrast to a regular set, allows for multiple occurrences of the same element. This concept is important in the context of permutations because it affects how you count possible arrangements.
When working with a multiset, such as the word "aloha," which contains repeated letters, special permutation formulas are needed. With multisets, we use a modified permutation formula to account for repeated items. The formula is: \[ \frac{n!}{n_1! \times n_2! \times \, ... \, \times n_k!} \], where \( n \) is the total number of elements, and \( n_1, n_2, \ldots \) are the frequencies of each repeating element.
For the word "aloha," where the letter 'A' appears twice, we calculate \( \frac{5!}{2!} \) to find the number of unique permutations where the indistinguishable 'A's do not simply swap places, resulting in 60 distinct arrangements.

Arrangement of Letters

Arranging letters, especially in a word like "aloha," involves understanding how different combinations can present themselves. Here, we want to know the number of distinct sequences we can form with these given letters.
Given that some letters repeat, traditional permutation formulas need to be adjusted for accuracy. Without repetition, the arrangement would be straightforward: simply calculate \( n! \), where \( n \) is the number of letters. However, because 'A' is repeated, we adjust using the multiset formula to ensure each arrangement is unique and distinct.
This adjustment is crucial; without it, identical sequences would be counted multiple times, skewing the results. Therefore, arranging letters with repetition accurately reflects the task’s complexity, providing the correct number of unique ways to form the word.