In how many distinct ways can the letters of the word TOOTH be arranged? ways
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Question:**
In how many distinct ways can the letters of the word TOOTH be arranged?
[Input box]
**Solution:**
To find the number of distinct arrangements of the letters in the word "TOOTH," we first note that it consists of 5 letters, where T appears twice and O appears twice. The formula for finding permutations of a multiset (where some items are repeated) is:
\[
\frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}
\]
Where:
- \( n \) is the total number of letters,
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the repeated letters.
For TOOTH:
- Total letters, \( n = 5 \).
- Frequency of T, \( n_1 = 2 \).
- Frequency of O, \( n_2 = 2 \).
- Frequency of H, \( n_3 = 1 \).
Applying the formula:
\[
\frac{5!}{2! \times 2! \times 1!} = \frac{120}{4} = 30
\]
Therefore, the letters of the word "TOOTH" can be arranged in 30 distinct ways.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0615f7c-3e91-45ec-8ebd-43a498fc758f%2Fd04e753c-1a46-4462-b203-23ac60c62f6e%2Feon3s9l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
In how many distinct ways can the letters of the word TOOTH be arranged?
[Input box]
**Solution:**
To find the number of distinct arrangements of the letters in the word "TOOTH," we first note that it consists of 5 letters, where T appears twice and O appears twice. The formula for finding permutations of a multiset (where some items are repeated) is:
\[
\frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}
\]
Where:
- \( n \) is the total number of letters,
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the repeated letters.
For TOOTH:
- Total letters, \( n = 5 \).
- Frequency of T, \( n_1 = 2 \).
- Frequency of O, \( n_2 = 2 \).
- Frequency of H, \( n_3 = 1 \).
Applying the formula:
\[
\frac{5!}{2! \times 2! \times 1!} = \frac{120}{4} = 30
\]
Therefore, the letters of the word "TOOTH" can be arranged in 30 distinct ways.
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