Unlocking the Secrets of Word Permutations

Table of Contents

1. Introduction
2. Understanding Permutations
3. Basic Concept of Factorial
4. Counting Letters in the Word
5. Identifying Duplicate Letters
6. Calculating Distinct Permutations
7. Simplifying the Computation
8. The Result: Distinct Permutations of "tallahassee"
9. Conclusion
10. References

Understanding Permutations

Permutations are a way to determine the number of possible arrangements or orderings of a set of elements. In this article, we will focus on calculating the number of distinct permutations that can be formed using the letters of a given word.

Basic Concept of Factorial

To calculate permutations, we need to understand the concept of factorial. Factorial is denoted by the exclamation mark (!) and represents the product of all positive integers up to a given number. For example, 5! is equal to 5 × 4 × 3 × 2 × 1, which is 120.

Counting Letters in the Word

The first step in determining distinct permutations is to count the number of letters in the word. For instance, in the word "tallahassee," we have a total of 11 letters.

Identifying Duplicate Letters

Next, we need to identify any duplicate letters in the word. These duplicate letters can affect the number of distinct permutations. Let's take a look at the word "tallahassee" and identify the duplicate letters:

• 'a' appears three times
• 'l' appears twice
• 's' appears twice
• 'e' appears twice

Calculating Distinct Permutations

To account for duplicate letters, we need to include factorials in the denominator of our permutation calculation. For each duplicate letter, we place the factorial of its frequency in the denominator. In the case of "tallahassee," the denominators would be:

• 3! (for 'a')
• 2! (for 'l')
• 2! (for 's')
• 2! (for 'e')

Simplifying the Computation

Now that we have the numerator (11!) and the denominator (3! × 2! × 2! × 2!), we can simplify the computation. Using a calculator, we calculate 11!, which equals 39916800, and the denominator, which equals 48.

The Result: Distinct Permutations of "tallahassee"

By dividing the numerator by the denominator, we find that the number of distinct permutations that can be formed using the letters of the word "tallahassee" is 831,600.

Conclusion

Calculating distinct permutations is essential when dealing with arrangements of letters or objects. By understanding the concept of factorial and accounting for duplicate letters, we can accurately determine the number of distinct permutations possible. In the case of "tallahassee," we found that there are 831,600 distinct permutations.

References

[1] "Factorial." Wikipedia. [2] "Permutations." Math is Fun.