jmc

First we count the number of all 4-letter words with no restrictions on the word. Then we count the number of 4-letter words with no vowels. We then subtract to get the answer.

Each letter of a word must be one of A, B, C, D, or E, so the number of 4-letter words with no restrictions on the word is $5\times 5\times 5\times 5=625$. Each letter of a word with no vowel must be one of B, C, or D. So the number of all 4-letter words with no vowels in the word is $3\times 3\times 3\times 3=81$. Therefore, the number of 4-letter words with at least one vowel is $625-81=\boxed{544}$.