Problem 25 - Algebra 2 and Trigonometry

Question

In \(23-30,\) find the number of different arrangements that are possible for the letters of each of the following words. EXCEED

Step-by-Step Solution

Answer

There are 120 different arrangements of the letters in "EXCEED".

Step 1: Determine the Total Number of Letters

The word "EXCEED" consists of 6 letters. To find the total number of possible arrangements of the letters, first note the total number of letters, which is 6.

Step 2: Calculate Factorial of Total Letters

The total number of ways to arrange 6 different letters is given by the factorial of 6, which is written as 6!. This represents the permutations of the letters if all were unique.

Step 3: Identify Repeated Letters

In the word "EXCEED", the letter 'E' appears 3 times. When letters are repeated, some arrangements will be identical and thus need to be divided out of the total permutations.

Step 4: Adjust for Repeated Letters

To account for the repeated letters, divide the total number of permutations by the factorial of the number of repeated letters. The formula for the number of distinct permutations is: \[ \frac{6!}{3!} \] where 3! accounts for the three 'E's.

Step 5: Calculate Factorials

First, calculate the factorials needed: \(6! = 720\) and \(3! = 6\).

Step 6: Find the Number of Unique Arrangements

Divide the total permutations by the permutations of the repeated letters: \( \frac{720}{6} = 120 \). Thus, there are 120 distinct arrangements of the letters in "EXCEED".

Key Concepts

Factorial Calculation

In mathematics, a factorial provides a way to calculate the number of permutations of a set of elements. The factorial of a number, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, to calculate the factorial of 6 (written as \( 6! \)), you multiply all integers from 1 up to 6: \( 6! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 = 720 \).
Factorials grow very quickly, which makes them a powerful tool for calculating permutations and combinations. They are used in various fields including mathematics, computer science, and probability theory.
When solving a permutation problem, the concept of a factorial helps in determining how many unique ways elements can be arranged if each element is different. It provides a baseline before further adjustments, like considering repeated elements, are made.

Distinct Arrangements

The concept of distinct arrangements comes into play when we want to know how many unique ways letters can be arranged in a word or elements in a set. In a simple case where all elements are different, the total number of distinct permutations is given by calculating the factorial of the total number of elements.
In our example with the word "EXCEED," if all six letters were different, the total number of distinct arrangements would be \( 6! = 720 \). However, not all letters are distinct due to repeated 'E's, which leads to identical permutations.

• This makes it necessary to adjust our calculation to only count unique arrangements.

The initial count using factorial assumes each letter is different, but in reality, some permutations will look the same, thanks to repeated letters. Thus, this calculation requires adjustment to eliminate duplicate patterns.

Repeated Letters Adjustment

Adjusting for repeated letters is essential to ensure the count of permutations only includes unique arrangements. When dealing with repetition, some arrangements of letters will be indistinguishable from others.
With the word "EXCEED," the letter 'E' appears 3 times. If calculated normally using \( 6! \) permutations, those indistinguishable arrangements get counted multiple times.
To correct this, we divide the total number of permutations by the factorial of the number of repeated letters: \[ \frac{6!}{3!} \].

• Here, \( 3! \) equals 6, representing all permutations formed by the three 'E's.
• By dividing, we remove these redundant arrangements from the total count.

This ensures we're counting only unique sequences, leading to the actual number of distinct arrangements, which in this case is 120. This step is crucial to accurately solve permutation problems involving repeated elements.