How Many Five-Letter Words Can Be Formed from Acme Vowel Soup?

How many five-letter words can be formed from a bowl of Acme Vowel Soup?

Acme Corporation has released an alphabet soup in which each of the vowels (A, E, I, O, U) of the English alphabet appears five times. How many different five-letter words can be created from this vowel soup?

Answer:

There are 3,125 different five-letter words that can be created from the given alphabet soup.

In the given alphabet soup, each vowel (A, E, I, O, U) appears five times, and the consonants do not appear at all. This means we have a total of 25 vowels (5 vowels * 5 appearances each).

To calculate the number of possible five-letter words that can be formed, we look at the concept of permutations. A five-letter word has five positions to fill. In each position, we can choose any of the five vowels available in the soup.

Using the formula for permutations, we find that the total number of possible arrangements is calculated as [tex]5^5[/tex]. This is because we have five positions to fill, and in each position, we have five choices (the five vowels) available.

Therefore, the number of different five-letter words that can be formed from the Acme Vowel Soup is [tex]5^5[/tex] = 3,125. Thus, there are 3,125 unique five-letter words that can be created from the given alphabet soup.

It's fascinating to see how these permutations allow us to explore the numerous possibilities that can arise from a seemingly simple set of letters. The concept of permutations is a powerful tool in combinatorial mathematics, offering insights into the arrangements and combinations that can be formed from a given set of elements.