How Many Different Ways Can 2 Letters from the Word 'Select' Be Arranged?

When it comes to combinatorial mathematics, determining the number of ways to arrange a given set of letters can be both intriguing and challenging. This article explores the permutations of 2 letters from the word 'select', a process that involves a series of steps to find the unique arrangements, including both identical and distinct letters.

Understanding the Problem

The word 'select' contains the letters s, e, l, c, t (5 unique letters with the letter 'e' appearing twice). The goal is to calculate the number of different ways to arrange any 2 letters from this word.

Step-by-Step Solution

Step 1: Count Unique Letters

The unique letters in 'select' are s, e, l, c, t, giving us a total of 5 unique letters. The recurrence of the letter 'e' is helpful to account for in the different scenarios.

Step 2: Consider Cases Based on the Letters Chosen

Choosing 2 Different Letters

When we choose 2 different letters from the 5 unique letters, we can use combinations to determine the number of possible pairs, which is given by:

$binom{5}{2} 10$

Each pair of different letters can be arranged in 2! (2 factorial) ways, which equals 2. Therefore, the total number of arrangements for this case is:

$10 times 2 20$

Choosing 2 Identical Letters

The only letter repeating in 'select' is 'e'. Therefore, the only way to choose 2 identical letters is to pick both 'e's, which can be arranged in only 1 way (ee).

Step 3: Add the Cases Together

Now, we add the arrangements from both cases mentioned above:

Arrangements with different letters: 20 Arrangements with identical letters: 1

Therefore, the total arrangements are:

$20 1 21$

Conclusion

The total number of different ways to arrange 2 letters from the word 'select' is 21. This approach ensures not only to account for all possible combinations but also to identify which arrangements are truly distinct.

Alternative Approach

Another method involves simplifying the problem by removing one of the 'e's from 'select', turning it into 'slect' or 'selct'. There are 5! / 3! 20 distinct permutations of 2 letters from 'slect' or 'selct'. Additionally, considering the initial scenario with two 'e's, we determine there are 30 total permutations, but 9 of them are not distinct. Subtracting these non-distinct permutations leaves us with 21 distinct arrangements.

Another Insight

If the selected 2 letters are not both 'e', any other pair of letters can be arranged in 2! 2 ways. For example, with 's' and 'l', we have 'sl' or 'ls'. This method reaffirms the conclusion of 21 distinct arrangements.

Understanding these steps not only helps in solving similar combinatorial problems but also provides insight into the underlying mathematical principles of permutations and combinations.