Example 5 How many ways you can arrange the letters in the word SCRAPED, if the first letter is not vowel and repetition of letters is allowed? n Counting Principle Permutation Notation Answer
Example 5 How many ways you can arrange the letters in the word SCRAPED, if the first letter is not vowel and repetition of letters is allowed? n Counting Principle Permutation Notation Answer
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![## Example 5
**Problem:**
How many ways can you arrange the letters in the word "SCRAPED", if the first letter is not a vowel and repetition of letters is allowed?
**Explanation:**
1. **Identifying Non-Vowel Letters:**
In the word "SCRAPED", the vowels are A and E, leaving the non-vowel letters as S, C, R, P, and D.
2. **Permutations with Repetition:**
Since repetition of letters is allowed, each position in the arrangement can be filled by any of the available letters (including vowels after the first position).
3. **Step-by-Step Process:**
- **First Letter:** Choose from the non-vowel letters (5 options: S, C, R, P, D).
- **Subsequent Letters:** Each of the remaining positions can be filled with any of the 7 letters in "SCRAPED".
4. **Using Permutation Notation and Counting Principle:**
- Let \( n \) be the number of available letters,
- Let \( r \) be the length of the word.
- The formula is given as \( n^r \) for each position.
\[
\text{Total arrangements} = 5 \times 7^6
\]
**Counting Principle Answer:**
Calculate the total number of arrangements accordingly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa34ee33c-a550-4b2b-8d8e-fb72d175c88c%2F3472ac6d-fe9d-42a3-a23f-2c4ef0eb335e%2F9666h2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Example 5
**Problem:**
How many ways can you arrange the letters in the word "SCRAPED", if the first letter is not a vowel and repetition of letters is allowed?
**Explanation:**
1. **Identifying Non-Vowel Letters:**
In the word "SCRAPED", the vowels are A and E, leaving the non-vowel letters as S, C, R, P, and D.
2. **Permutations with Repetition:**
Since repetition of letters is allowed, each position in the arrangement can be filled by any of the available letters (including vowels after the first position).
3. **Step-by-Step Process:**
- **First Letter:** Choose from the non-vowel letters (5 options: S, C, R, P, D).
- **Subsequent Letters:** Each of the remaining positions can be filled with any of the 7 letters in "SCRAPED".
4. **Using Permutation Notation and Counting Principle:**
- Let \( n \) be the number of available letters,
- Let \( r \) be the length of the word.
- The formula is given as \( n^r \) for each position.
\[
\text{Total arrangements} = 5 \times 7^6
\]
**Counting Principle Answer:**
Calculate the total number of arrangements accordingly.
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