### 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, I can use the complement principle and subtract the number of 4-letter words with no consonants from the total number of 4-letter words.
The total number of 4-letter words is simply $5^4$, since each letter has 5 choices and the order matters.
The number of 4-letter words with no consonants is just the number of ways to choose 4 letters from the 2 vowels $A$ and $E$, which is $2^4$.
Therefore, the number of 4-letter words with at least one consonant is $5^4 - 2^4$.