Question

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 (and the consonants do not appear at all). How many five-letter words can be formed from a bowl of Acme Vowel Soup? (Note: The words do not have to be actual words in English!)

Answer 1

Given that the alphabet soup contains the vowels (A, E, I, O, U) with each of the vowels appearing five times.

This means that the total number of letters in the bowel soup is 25 letters.

The number of five-letter words that can be formed from the bowel of 25 letters assuming there are no restrictions is given by
$^{25}P_5= \frac{25!}{(25-5)!} \\ \\ = \frac{25!}{20!} =25\times24\times23\times22\times21 \\ \\ =6,375,600$