SOLUTION: In the ordinary alphabet of 26-letters. a) Define a "4-letter word" to be any list of 4 letters that contains at least one of the vowels A, E, I, O, U. How many 4-letter words a

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Question 304515: In the ordinary alphabet of 26-letters.
a) Define a "4-letter word" to be any list of 4 letters that contains at least one of the vowels A, E, I, O, U. How many 4-letter words are there?
b) Suppose, instead, we define a "4-letter word" to be any list of 4 letters that contains exactly one of the vowels A, E, I, O, U. How many 4-letter words are there?
Found 2 solutions by vleith, stanbon:
Answer by vleith(2983) (Show Source):
You can put this solution on YOUR website!
Assume a four letter word
If the first letter is a vowel, then it can be 1 of 5 possible letters.
The remaining 3 letters in the word can be any of 26 letters. So
gives a word with the first letter as a vowel.
gives a word with the second letter as a vowel.
gives a word with the third letter as a vowel.
gives a word with the fourth letter as a vowel.
The total would be thge sum of those four products.

B)
Part B is a misleading question. Does it mean there is a single vowel in the word or does it mean that multiple occurrences of the same vowel is allowed?
If a single vowel only is allowed then
If the first letter is a vowel, then it can be 1 of 5 possible letters.
The remaining 3 letters in the word can be any consonant letters. So
gives a word with the first letter as a vowel.
gives a word with the second letter as a vowel.
gives a word with the third letter as a vowel.
gives a word with the fourth letter as a vowel.
The total would be thge sum of those four products.
If you are allowed multiple occurrences of the same vowel, then change the 21 above to 22 (since you can have the same vowel again)

Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
In the ordinary alphabet of 26-letters.
a) Define a "4-letter word" to be any list of 4 letters that contains at least one of the vowels A, E, I, O, U. How many 4-letter words are there?
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Assuming that repetition is not allowed:
There are 26*25*24*23 4-letter words with no restrictions.
There are 19*18*17*16 4-letter words that have no vowels.
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So, there are (26*25*24*23)-(19*18*17*16) 4-letter words that have at least one vowel.
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b) Suppose, instead, we define a "4-letter word" to be any list of 4 letters that contains exactly one of the vowels A, E, I, O, U. How many 4-letter words are there?
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5-ways to choose the vowel
19*18*17 ways to choose the other three letters
Total words = 5*19*18*17
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Cheers,
Stan H.