How many ways can the letters of a word be rearranged using Python?

Determining how many ways the letters of a word can be rearranged is a classic problem in combinatorics, often involving factorials. Python makes it relatively easy to calculate this, especially when dealing with words with repeating characters.

Here's how to approach it:

1. Factorial for Unique Characters: If all the letters in the word are unique, then the number of rearrangements is simply the factorial of the number of letters. For example, the word 'CAT' has 3 unique letters, so the number of rearrangements is 3! = 3 2 1 = 6.

2. Handling Repeating Characters: If the word contains repeating characters, we need to divide the factorial by the factorials of the count of each repeating character. For example, the word 'AAB' has two 'A's. The total number of arrangements without considering repetition would be 3! = 6. However, since the two 'A's are identical, we need to divide by 2! (the factorial of the number of times 'A' is repeated) giving 3! / 2! = 6 / 2 = 3 arrangements.

3. Using Python's `math` Module: Python’s math module has a factorial function that is very helpful. We can use the Counter object from the collections module to count character occurrences.

4. Example Python Code:

import math
from collections import Counter

def count_rearrangements(word):
    word_length = len(word)
    char_counts = Counter(word)
    numerator = math.factorial(word_length)
    denominator = 1
    for count in char_counts.values():
        denominator = math.factorial(count)
    return numerator // denominator

word1 = "CAT"
word2 = "AAB"
word3 = "MISSISSIPPI"

print(f"The word '{word1}' can be rearranged in {count_rearrangements(word1)} ways.")
print(f"The word '{word2}' can be rearranged in {count_rearrangements(word2)} ways.")
print(f"The word '{word3}' can be rearranged in {count_rearrangements(word3)} ways.")


This Python code calculates the number of rearrangements effectively, handling both unique and repeating characters.

5. How the Code Works:

- The `count_rearrangements` function first calculates the factorial of the word's length.

- It uses `Counter` to tally up how often each letter appears.

- Then, it multiplies together the factorials of the character counts.

- Finally, it divides the word length's factorial by the product of the factorials of each letter's counts, effectively removing overcounted arrangements due to repetitions.

This approach ensures accurate and fast computation of all possible letter rearrangements for any given word in Python.

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