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How many 4-letter words with at least one vowel can be constructed from the letters A, B, C, D, and E? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)

 Sep 3, 2018

For this problem it's probably easiest to determine the number of 4 letter words with no vowels, and subtract that from the total number of possible words.


The number of words with no vowels is simply \(3^4\)

(do you see why?)


The total number of possible words is \(5^4\)


so the total number of words with at least 1 vowel is


\(n = 5^4 - 3^4 = 544\)


I suppose as a check we can do it the hard way.  The number will be the sum of all words with 1, 2, 3, and 4 vowels.  I.e.


\(2^1 \dbinom{4}{1}3^3+2^2 \dbinom{4}{2}3^2+2^3 \dbinom{4}{3} 3^1 + 2^4 = 544\)

 Sep 3, 2018

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