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 d, and are consonants, any word is valid, not just English language words, and letters may be used more than once.)
4 letter words with 4 consonants = Any of 3 consonants can occupy all 4 positions = (3)^4 = 81 words
4 letter words with 3 consonants = Any of the 3 consonants can be chosen for any 3 of 4 positions and we have 2 choices for the final position of the vowel....so (3)^3 * 4C3 * 2 = 216 words
4 letter words with 2 consonants = Any of the 3 consonants can be chosen for 2 of the 4 positions and we have 2^2 ways to choose the vowels for the other 2 postions = (3)^2 * 4C2 * 2^2 = 216 words
4 letter words with 1 consonant = And of the 3 consonants can be chosen for 1 of the four positions and we have (2)^3 ways to choose the vowels for the other 3 positions = (3) * 4C1 * 2^3 = 96 words
81 + 216(2) +96 = 609 words
Another way to see this is that we have (5)^4 = 625 possible words
And the number of words that contain no consonants = (2)^4 = 16
So...the number of words with at least one consonant = 625 - 16 = 609 words
The dictionary defines WORD as "a single distinct meaningful element of speech or writing, used with others (or sometimes alone) to form a sentence and typically shown with a space on either side when written or printed."
By this definitioin, something like BCBD is not a word.
The question should be "How many 4-letter
words character strings ... can be constructed..."
in my humble opinion
Yes, but the question noted "any word is valid, not just English language words, and letters may be used more than once" so it had the meaning of "character string"
Most people won't recognize what a character string is so they worded the problem like that.