Problem 39 - College Algebra with Corequisite Support

Question

How many arrangements can be made from the letters of the word "mountains" if all the vowels must form a string?

Step-by-Step Solution

Answer

There are 8,640 ways to arrange the letters in 'mountains' with all vowels forming a string.

Step 1: Identify the Components

The word 'mountains' consists of 9 letters. The vowels in this word are 'o', 'u', 'a', and 'i'. We need to treat all vowels as a single unit or "block" for the purpose of this problem. This means we can consider the block of vowels as a single "letter" initially.

Step 2: Calculate Vowel Arrangements

Within the block of vowels, the vowels can be arranged amongst themselves. There are 4 vowels, and they can be arranged in any order. This can be calculated by finding the factorial of the number of vowels: \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Step 3: Calculate Total Arrangements

Considering the vowel block as one "letter," we now have the rest of the letters: 'm', 'n', 't', 'n', and 's'. Including the vowel block, we have 6 "letters" to arrange. Since 'n' repeats twice, the total arrangements are computed by \( \frac{6!}{2!} = \frac{720}{2} = 360 \).

Step 4: Calculate Final Arrangements

To find the total number of arrangements of the word 'mountains', we multiply the number of arrangements of the vowel block with the number of arrangements of all blocks: \((24) \times (360) = 8640\).

Key Concepts

Factorial

The concept of factorial is crucial when calculating permutations. It's essentially the product of all positive integers up to a certain number. For example, the factorial of 4, denoted by \(4!\), is calculated by multiplying 4 by every whole number smaller than it, down to 1:

• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

A factorial helps determine how many ways a set of items, such as letters, can be arranged in different sequences. The larger the number, the more complex and numerous the possible arrangements. This principle underlies much of combinatorial mathematics as it helps in counting permutations where the order of items matters.

Repeated Letters

In permutations, repeated letters alter the total number of unique arrangements possible. Consider the word "mountains," where the letter 'n' appears twice. If all the letters were unique, you could simply use the factorial of the total count of letters to find the permutations. However, when repetitions occur, you must adjust by dividing by the factorial of the number of times each repeated letter appears.Here's how it works:

• Identify the total number of "letters", considering unique elements and blocks (like the vowel block in our case).
• Adjust for repetitions by dividing the total factorial by the factorial of the count of each repeated letter. Here, \(\frac{6!}{2!}\) accounts for the duplicated 'n'.

This adjustment ensures that identical sequences aren't counted multiple times, leading to an accurate count of distinct permutations.

Vowel Block

When solving permutation problems, the concept of treating a group of vowels as a single entity or "block" helps simplify calculations. In the example of the word "mountains," vowels 'o', 'u', 'a', and 'i' are grouped as one block.This step reduces the complexity of the arrangement problem:

• Consider all vowels as a single "letter" reducing the number of elements to arrange.
• Internally, rearrange only the vowels to consider all possible sequences within the block.

The permutations of the vowel block are calculated separately (here, \(4!\) for 4 vowels), and then this permutation count is multiplied with the permutations of the entire group of letters and blocks.

This approach is efficient as it breaks down a more complex arrangement into a series of manageable calculations.

Combinatorics

Combinatorics is the area of mathematics primarily concerned with counting, arranging, and combining items within set rules. It's a key concept for understanding permutation problems like those found in the "mountains" exercise. The main steps involved in combinatorics for this problem are:

• Calculate possible arrangements for defined groups or blocks (like the vowel block).
• Consider any restrictions, such as repeated letters, and adjust the permutations accordingly.
• Multiply results from each stage of the problem to find the total number of permutations.

Understanding combinatorics allows students to solve problems efficiently by using structured approaches to rearrange and count items, which can be translated into many real-world applications, from schedule planning to DNA sequencing.