Distinct Letter Arrangements: Nappanee & Other City Names

by Alex Johnson 58 views

Have you ever wondered how many different ways you can arrange the letters in a word? It's a fascinating problem in mathematics called permutations, and it gets even more interesting when some letters are repeated! Let's dive into how we can calculate the number of distinct arrangements, or permutations, for the letters in several city names.

Understanding Permutations

Before we jump into our city names, let's quickly review the basics of permutations. A permutation is an arrangement of objects in a specific order. If all the objects are distinct, the number of permutations of n objects is simply n! (n factorial), which means n × (n-1) × (n-2) × ... × 2 × 1. For example, the number of ways to arrange the letters in the word "CAT" is 3! = 3 × 2 × 1 = 6. However, when we have repeated letters, we need to adjust our calculation to avoid counting the same arrangement multiple times.

The key concept here is understanding that repeated letters create identical arrangements if swapped. To correct for this overcounting, we divide the total factorial by the factorial of the count of each repeated letter. This ensures that we only count truly distinct arrangements.

Imagine you have the word "APPLE". There are 5 letters, so initially, you might think there are 5! = 120 ways to arrange them. But the letter 'P' is repeated twice. Swapping the two 'P's doesn't create a new arrangement. Therefore, we divide by 2! (which is 2 × 1 = 2) to account for the repeated 'P'. So, the distinct arrangements for "APPLE" are 5! / 2! = 120 / 2 = 60. This principle forms the basis for solving all the city name arrangements below, allowing us to accurately determine the number of unique permutations for each.

1. Nappanee

Let's start with Nappanee. This city name has 8 letters. If all letters were unique, we would have 8! arrangements. However, the letter 'P' appears twice, the letter 'A' appears twice, and the letter 'N' appears twice. To find the number of distinct arrangements, we use the formula:

Number of arrangements = 8! / (2! × 2! × 2!)

Let's break this down:

  • 8! (8 factorial) = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320
  • 2! (2 factorial) = 2 × 1 = 2

So, our calculation becomes:

40,320 / (2 × 2 × 2) = 40,320 / 8 = 5,040

Therefore, there are 5,040 distinct ways to arrange the letters in Nappanee. This calculation effectively demonstrates how accounting for repeated letters significantly reduces the total number of permutations, providing an accurate count of unique arrangements. Without this adjustment, we would greatly overestimate the number of distinct possibilities.

2. Rensselaer

Next, let's tackle Rensselaer. This name has 10 letters. The letter 'E' appears twice, and the letter 'R' appears twice. The rest of the letters appear only once. Using the same logic as before, the number of distinct arrangements is:

Number of arrangements = 10! / (2! × 2!)

Let's calculate this:

  • 10! (10 factorial) = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800
  • 2! (2 factorial) = 2 × 1 = 2

So, the calculation is:

3,628,800 / (2 × 2) = 3,628,800 / 4 = 907,200

Thus, there are a whopping 907,200 different ways to arrange the letters in Rensselaer. The significant increase in permutations compared to Nappanee highlights how the number of letters and their repetition rates exponentially affect the final result. Even a slight increase in the word length can lead to dramatic differences in the number of possible arrangements.

3. Sassafras

Now, let's consider Sassafras. This word contains 9 letters, with 'S' appearing four times and 'A' appearing three times. To determine the number of unique arrangements, we'll use the formula that accounts for these repetitions:

Number of arrangements = 9! / (4! × 3!)

Here's the breakdown:

  • 9! (9 factorial) = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362,880
  • 4! (4 factorial) = 4 × 3 × 2 × 1 = 24
  • 3! (3 factorial) = 3 × 2 × 1 = 6

Therefore, the calculation is:

362,880 / (24 × 6) = 362,880 / 144 = 2,520

This means there are 2,520 distinct ways to arrange the letters in Sassafras. This lower number compared to Rensselaer, despite having nearly the same number of letters, underscores the significant impact that a higher frequency of repeated letters has on reducing the number of unique arrangements. The more letters that are repeated, the fewer distinct permutations are possible.

4. Summitville

Moving on to Summitville, we have 10 letters. The letter 'M' appears twice, and the letter 'T' appears twice. The formula for distinct arrangements is:

Number of arrangements = 10! / (2! × 2!)

Let's calculate:

  • 10! (10 factorial) = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800
  • 2! (2 factorial) = 2 × 1 = 2

So, the equation is:

3,628,800 / (2 × 2) = 3,628,800 / 4 = 907,200

As a result, there are 907,200 different arrangements of the letters in Summitville. This is the same number of arrangements as Rensselaer, highlighting how different letter combinations with the same repetition patterns can result in the same number of distinct permutations.

5. Uniontown

Let's analyze Uniontown, which has 9 letters. The letter 'N' appears twice, and the letter 'O' appears twice. The calculation for distinct arrangements is:

Number of arrangements = 9! / (2! × 2!)

Here's how we break it down:

  • 9! (9 factorial) = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362,880
  • 2! (2 factorial) = 2 × 1 = 2

Therefore, the calculation is:

362,880 / (2 × 2) = 362,880 / 4 = 90,720

Thus, there are 90,720 unique ways to arrange the letters in Uniontown. This number is significantly lower than Summitville and Rensselaer, despite having a similar number of letters, due to the specific pattern of letter repetitions within the word.

6. Wawasee

Now we'll calculate for Wawasee, which has 7 letters. The letter 'A' appears twice, and the letter 'W' appears twice, and the letter 'E' appears twice. The formula for distinct arrangements is:

Number of arrangements = 7! / (2! × 2! × 2!)

Let's break it down:

  • 7! (7 factorial) = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040
  • 2! (2 factorial) = 2 × 1 = 2

So, the calculation is:

5,040 / (2 × 2 × 2) = 5,040 / 8 = 630

This means there are 630 distinct ways to arrange the letters in Wawasee. This relatively low number, compared to other city names, is a direct result of the high frequency of repeated letters in the word, which significantly reduces the possible unique permutations.

7. Carpentersville

Finally, let's consider Carpentersville, which boasts a substantial 15 letters. The letter 'E' appears three times, the letter 'R' appears twice, and the letter 'P' appears twice. To calculate the number of distinct arrangements, we apply the following formula:

Number of arrangements = 15! / (3! × 2! × 2!)

Here's the detailed calculation:

  • 15! (15 factorial) = 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 1,307,674,368,000
  • 3! (3 factorial) = 3 × 2 × 1 = 6
  • 2! (2 factorial) = 2 × 1 = 2

So, the calculation is:

1,307,674,368,000 / (6 × 2 × 2) = 1,307,674,368,000 / 24 = 54,486,432,000

Therefore, there are an astounding 54,486,432,000 distinct ways to arrange the letters in Carpentersville. This massive number underscores how the factorial function grows rapidly with the number of letters, and even with repetitions, longer words have a vast number of potential arrangements. The sheer magnitude illustrates the complexity that arises when dealing with permutations of larger sets.

Conclusion

Calculating the number of distinct arrangements of letters is a wonderful exercise in combinatorics. It highlights how repetition affects the number of possibilities and gives us a deeper appreciation for mathematical principles in everyday words and names. We've seen how seemingly simple words can have thousands, even billions, of different arrangements! To further explore the fascinating world of permutations and combinations, consider visiting resources like Khan Academy's Combinatorics section.