SOLUTION: How many 4-letter words with at least one vowel can be constructed from the letters A, B, C, D, and E?

Question 390475: How many 4-letter words with at least one vowel can be constructed from the letters A, B, C, D, and E?
Found 3 solutions by haileytucki, sudhanshu_kmr, richard1234:
Answer by haileytucki(390)   (Show Source): You can put this solution on YOUR website!
There are 5 letters and the question asks for How many four letter words?
This is a possibility problem:
5P4
Find the number of possible ordered permutations when r items are selected from n available items.
5P4=nPr=(n!)/(n-r!)
Fill in the known values.
(5!)/((5-4)!)
Cancel out the common factorial factors.
5*4*3*2
Multiply 5 by 4 to get 20.
20*3*2
Multiply 20 by 3 to get 60.
60*2
Multiply 60 by 2 to get 120.
Answer= 120
Answer by sudhanshu_kmr(1152)   (Show Source): You can put this solution on YOUR website!
Here there r 5 letters in which 2 are vowels and 3 consonants.
thus, it is not possible to form 4 letter words without vowel, so condition of at least one vowel will always satisfy...

no. of possible words = 5P4 = 120
Answer by richard1234(7193)   (Show Source): You can put this solution on YOUR website!
The total number of four-letter "words" that can be made (duplicating letters if necessary) is 5^4, or 625, since for each position, any of the five letters can appear there.

We wish to count the number of four letter "words" with no vowels. Then, we can subtract this number from 625 to obtain the number of words with at least one vowel. The number of four letter words with no vowels is 3^4 or 81, since a B, C, or D can appear in any position.

Therefore the number of four letter words that satisfy is equal to 5^4 - 3^4, or 544.

The case where each letter in the word must be distinct is slightly different. If each letter is distinct, then you are guaranteed to have at least one vowel in the word. Therefore the answer would be 5P4, or 5*4*3*2, or 120. I do not know if the problem asks for distinct letters or not, it could be either 544 or 120 depending on the case.
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