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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 B, C, and D are consonants, any word is valid, not just English language words, and letters may be used more than once.)

 Oct 29, 2020
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A, B, C, D, E. Normally, 5 letters taken 4 at a time, you would have:5P4 =120 permutations. But, because each letter can be repeated up to 4 times and each of the 4 letters has 5 options, then the total number of permutations is simply: 5^4 =625  four-letter "words".

 

However, since you want each 4-letter word to have at least one vowel in it(A, or E), then we have to figure out those permutations that don't have at least one vowel in them. You have 3 consonants, B, C, and D and since each can be repeated up to 4 times, we therefore have: 3^4 =81 permutations which will have NO vowels in them. We, therefore, have to subtract them from the above total of 625 permutations:

 

So, 625 - 81 =544 four-letter "words" which have at least one vowel in them.

 Oct 30, 2020

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