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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
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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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