Problem 2 - Discrete Mathematics with Applications

Question

Find the number of distinct words that can be formed by scrambling the letters in each word. TALLAHASSEE

Step-by-Step Solution

Answer

The number of distinct words that can be formed by scrambling the letters in the word "TALLAHASSEE" is 415800.

Step 1: Identify the given word and count the letters in it

The given word is "TALLAHASSEE," and it has 11 letters in total.

Step 2: Count the occurrences of each distinct letter

We need to calculate the number of occurrences of each distinct letter. - T occurs 1 time - A occurs 4 times - L occurs 2 times - H occurs 1 time - S occurs 2 times - E occurs 1 time

Step 3: Calculate the total permutations

To calculate the total permutations, we will use the formula: \[n! = \frac{11!}{1!4!2!1!2!1!}\] Now, we need to calculate the factorials of all the numbers: \(11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1= 39916800\) \(1! = 1\) \(4! = 4 \times 3 \times 2 \times 1 = 24\) \(2! = 2 \times 1 = 2\) Calculate the permutations: \(n! = \frac{39916800}{1 \times 24 \times 2 \times 1 \times 2 \times 1} = \frac{39916800}{96} = 415800\)

Step 4: Find the number of distinct words

The number of distinct words that can be formed by scrambling the letters in the word "TALLAHASSEE" is 415800.

Key Concepts

Factorials

Understanding factorials is integral to solving permutation problems. A factorial of a non-negative integer is the product of all positive integers less than or equal to that number. It is denoted by an exclamation mark (!). For example, the factorial of 5 is calculated as follows:

• 5! = 5 × 4 × 3 × 2 × 1 = 120.

Factorials grow very quickly with larger numbers. For instance, 11! (11 factorial) results in a very large number, which is 39,916,800. When calculating permutations, you often divide one factorial by another, which is the case with our word problem involving the distinct word formation.
In the case of "TALLAHASSEE," we use factorials to arrange the total letters and divide by the factorials of each count of the repeated letters. This allows us to account for those repetitions and find the distinct arrangements. So here, while 11! gives us all possible arrangements of 11 letters, dividing by 4! for the letter "A," 2! for "L," and others helps ensure we only count distinct permutations. This is how we arrive at 415,800 distinct words.

Distinct Letter Counting

Counting distinct letters is crucial when dealing with permutations, particularly for words like "TALLAHASSEE" with repeating letters. To determine how many unique words can be created from our given word, we first need to identify each letter and count their occurrences.
In "TALLAHASSEE," each letter appears as follows:

• T: 1 time
• A: 4 times
• L: 2 times
• H: 1 time
• S: 2 times
• E: 1 time

These counts are then used in calculating permutations. Calculating distinct arrangements requires dividing the factorial of the total number of letters by the factorial of the count of each letter. Each factorial division reduces the possible arrangements, helping eliminate duplicates created by indistinguishable letter repetitions.
So in "TALLAHASSEE," the use of factorials of distinct letter counts allows us to compute accurate distinct permutations, ensuring no arrangement is doubly counted due to repetition.

Combinatorics

Combinatorics is the branch of mathematics principally concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. In simpler terms, it is about figuring out how many ways we can arrange or select things.
When dealing with problems like creating words from "TALLAHASSEE," combinatorics helps us handle the complexities introduced by repeated letters.

• It uses concepts like permutations in conjunction with factorials to calculate the total number of unique arrangements.
• It systematically accounts for repetitions by adjusting factorials appropriately for letters that appear more than once.

For our word problem, the main challenge is managing repeating letters. The combinatoric approach allows us to simplify the counting process through structured formulae. These formulae incorporate factorials to adjust for identical items, streamlining a manually impossible task.
This method is a powerful tool in mathematics and is widely used in various fields, including statistics, computer science, and optimization, to solve complex counting problems efficiently.