Exploring Combinatorics: Counting Words from Four Letters with Repetition

Combinatorics is a fundamental branch of mathematics that deals with counting, arrangement, and selection of objects. In this detailed guide, we will explore a specific problem involving the English alphabet: how many different words can be formed using four letters, with the condition that one letter can only be used an odd number of times and letters can be repeated. This exploration will involve a step-by-step approach and an understanding of combinatorial principles.

Understanding the Problem

The problem at hand requires us to form words from the English alphabet (26 letters) using exactly four letters, with one letter used an odd number of times. The two possible configurations are:

4 letters, where one letter is used an odd number of times, and the others are used an even number of times. 1 1 1 2, where one letter is used an odd number of times, and the remaining three letters are the same, used an even number of times.

Combinatorial Approach

Let's break down the problem into manageable steps:

Choose the letter that will be used an odd number of times from the 26 letters available. Determine the possible arrangements for the chosen letter in combination with the other three letters.

Step 1: Choosing the Odd Occurrence Letter

There are 26 choices for the letter that will be used an odd number of times. Let's denote this letter by L.

Step 2: Arranging the Letters

We need to consider two main scenarios:

Scenario 1: 4 letters where one letter is used an odd number of times, and the others are used an even number of times.

In this scenario, the only letter that appears an odd number of times is L. The other three letters can be any of the 25 remaining letters and must be used an even number of times, which means at least one of them will be eliminated in one of the positions. Hence, we have the following sub-scenarios:

3L and 1 other: 2L and 2 others 1L and 3 others

Let's calculate the permutations for each sub-scenario:

Sub-scenario 1: 3L and 1 other

Choose 1 letter from the remaining 25 letters: 25 choices.

Arrange 4 letters where 3 are L and 1 is another letter: 4 permutations.

Total permutations: 25 × 4 100.

Sub-scenario 2: 2L and 2 others

Choose 2 letters from the remaining 25 letters: 25 choose 2 300.

Arrange 4 letters where 2 are L and 2 others are different: 6 permutations.

Total permutations: 300 × 6 1800.

Sub-scenario 3: 1L and 3 others

Choose 1 letter from the remaining 25 letters: 25 choices.

Arrange 4 letters where 1 is L and 3 others are different: 4 permutations.

Total permutations: 25 × 4 100.

Total permutations for Scenario 1: 100 1800 100 2000.

Total permutations for both letters and their combinations: 26 × 2000 52000

Scenario 2: 1 1 1 2

In this scenario, one letter appears three times and another different letter appears once. Here, the letter that appears three times is our chosen letter L, and the other letter is any of the 25 remaining letters. We will calculate the permutations for this scenario as follows:

Choose 1 letter from the remaining 25 letters: 25 choices.

Arrange 4 letters where 3 are the same L and 1 is a different letter: 4 permutations.

Total permutations: 25 × 4 100.

Conclusion

Through this detailed exploration, we have determined that the total number of permutations for forming words with four letters from the English alphabet, where one letter can be used only an odd number of times and letters can be repeated, is 52,000. The problem involves considering various combinatorial scenarios and calculating their respective permutations.

Related Keywords

Combinatorics, Permutations, English Language, Word Combinations