Exploring 5-Letter Arrangements of the Word DISCOVERY

The word DISCOVERY is a nine-letter word containing nine distinct characters. This article explores the total number of unique 5-letter arrangements that can be formed using these letters. We will use both permutation and combination principles to calculate the possible arrangements, ensuring our calculations align with Google’s search engine optimization standards.

Understanding the Problem

The word DISCOVERY consists of nine unique letters: D, I, S, C, O, V, E, R, Y. Our goal is to determine how many different 5-letter arrangements can be made using these letters without repetition.

Step-by-Step Solution

We can approach this problem in a logical manner by dividing it into two cases: Case 1 where all 5 letters are different, and Case 2 where one letter appears twice and the other three are different. However, since the word DISCOVERY has only one of each letter, Case 2 is not applicable, and we only need to consider Case 1.

Case 1: All 5 Letters are Different

In this case, we need to choose 5 different letters from the 9 available and then arrange them.

Choosing 5 Letters

The number of ways to choose 5 letters from 9 is given by the combination formula binom{n}{r}, where n 9 and r 5.

Mathematically, this is represented as:

[ binom{9}{5} frac{9!}{5! times (9-5)!} frac{9!}{5! times 4!} ]

Calculating the factorial values, we get:

[ binom{9}{5} frac{9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1}{(5 times 4 times 3 times 2 times 1) times (4 times 3 times 2 times 1)} ]

which simplifies to:

[ binom{9}{5} frac{9 times 8 times 7 times 6}{4 times 3 times 2 times 1} 126 ]

Arranging the Chosen Letters

After choosing 5 letters, we can arrange them in different orders. The number of ways to arrange 5 different letters is given by the factorial of 5, i.e., 5!.

This is represented as:

[ 5! 5 times 4 times 3 times 2 times 1 120 ]

Total Arrangements for Case 1

Therefore, the total number of 5-letter arrangements for Case 1 is:

[ text{Total arrangements} binom{9}{5} times 5! 126 times 120 15120 ]

Conclusion

Since the word DISCOVERY has no repeating letters, the total number of unique 5-letter arrangements is 15120.

For clarity and verification, we can also use the permutation formula nPr frac{n!}{(n-r)!}, where n 9 and r 5.

This gives us:

[ 9P5 frac{9!}{(9-5)!} frac{9!}{4!} 15120 ]

Permutation and Combination Notation

The symbol 9P5 represents the number of ways to arrange 5 items out of 9, which is the same as frac{9!}{4!}. This notation simplifies the calculation by directly representing the total number of permutations without explicitly calculating the factorial terms.

Application in SEO and Google’s Standards

This method not only provides a clear and accurate solution but also adheres to SEO best practices by breaking down the problem into intuitive steps, using proper mathematical notation, and ensuring the content is well-structured with headings and paragraphs.

By utilizing these techniques, the content becomes more accessible to search engines like Google, enhancing its ranking and visibility in search results.