Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Use the fundamental principle of counting or permutations to solve each problem. How many ways can all the letters of the word ELTON be arranged?

Step-by-Step Solution

Answer

120 ways.

Step 1: Understanding the problem

The problem is asking for the number of possible arrangements (or permutations) of the letters in the word 'ELTON'. Since there are 5 distinct letters in 'ELTON', we'll be using permutations of distinct items.

Step 2: Applying the permutation formula

For any set of distinct items, the number of permutations is given by the factorial of the number of items. Here, we have 5 distinct letters, so we calculate the number of permutations as \(5!\) (5 factorial).

Step 3: Calculating 5 factorial

Calculate \(5!\) by multiplying the consecutive positive integers from 5 down to 1. That is, \(5! = 5 \times 4 \times 3 \times 2 \times 1\).

Step 4: Performing the multiplication

Calculate the multiplication: \(5 \times 4 = 20\); \(20 \times 3 = 60\); \(60 \times 2 = 120\); and \(120 \times 1 = 120\).

Step 5: Conclusion

The number of different ways to arrange the letters of 'ELTON' is 120.

Key Concepts

Fundamental Principle of Counting

The fundamental principle of counting is a key concept in permutations and combinations. It provides a simple way to find the number of possible outcomes in a sequence of events. Imagine you have multiple steps to complete, and each step has a set of options to choose from. The principle states that the total number of outcomes is the product of the options available at each step.

For example:

• If you have 3 shirts and 2 pairs of pants, the total number of outfits you can make is \(3 \times 2 = 6\).

In the case of arranging letters, each position in the sequence (first, second, etc.) counts as a step, and the number of letters left to choose from counts as the options. By multiplying these options, you determine the total number of arrangements (permutations). In the provided exercise, all 5 letters in "ELTON" are distinct, so the fundamental principle of counting guides us to calculate \(5!\).

Factorial

Factorials are a mathematical tool used to calculate the permutations of a set of items. The factorial of a number \(n\), written as \(n!\), is the product of all positive integers from \(n\) down to 1. It's a simple concept but immensely powerful when dealing with permutations.

Here's how it works:

• For \(3!\), calculate \(3 \times 2 \times 1 = 6\)
• For \(4!\), calculate \(4 \times 3 \times 2 \times 1 = 24\)
• For \(5!\), calculate \(5 \times 4 \times 3 \times 2 \times 1 = 120\)

The exercise uses this concept. To find the number of arrangements for 5 letters in "ELTON", you compute \(5!\), resulting in 120 possible permutations. Factorials scale quickly as \(n\) increases, making them an efficient tool for calculating large sets of arrangements and outcomes.

Distinct Items

When the concept of distinct items is mentioned, it refers to elements that are different from each other. Each item in the set is unique. In permutation problems, this distinction is crucial because it ensures that each item can occupy a unique position in an arrangement.

Consider the word "ELTON":

• It has 5 distinct letters: E, L, T, O, and N.
• Each of these letters can take any of the 5 positions in an arrangement.

Because each letter is different, we apply the permutation formula without any adjustment for repetition. If items were repeated, the calculation would involve dividing by the factorial of the number of identical items to account for indistinct permutations. Understanding that all items are distinct clarifies that the direct application of \(5!\) is appropriate, leading us to the 120 unique arrangements of the word "ELTON".

Arrangement of Letters

The arrangement of letters refers to the different ways letters can be sequenced in a given word. This is a typical application of permutations in mathematics. To find these arrangements, particularly when letters are distinct, you can apply the factorial of the number count of letters as a method of calculation.

For "ELTON":

• The number of letters is 5.
• You calculate the factorial of 5, which is \(5! = 120\).

This means there are 120 unique orders you can place the letters E, L, T, O, and N. Arranging distinct items like letters relies on the principles of permutations, which help enumerate every possible sequence. This is essential for both understanding combinations and solving problems in probability and combinatorics, providing insight into how different components can fit together in various patterns.