### Instruction: How many 4-letter words with at least one consonant can be constructed from the letters $A$, $B$, $C$, $D$, and $E$?  (Note that $B$, $C$, and $D$ are consonants, any word is valid, not just English language words, and letters may be used more than once.).### Response: To count the number of 4-letter words with at least one consonant, it might be easier to count the number of 4-letter words with no consonants and subtract that from the total number of 4-letter words.
The total number of 4-letter words is simply $5^4$, since there are 5 choices for each letter.
The number of 4-letter words with no consonants is $2^4$, since there are only 2 choices ($A$ or $E$) for each letter.
Therefore, the number of 4-letter words with at least one consonant is $5^4 - 2^4$.