TITLE: String of letters and ways to have at least one vowel
QUESTION [5 upvotes]: We would like to construct $3$ letter strings from the english alphabet ($26$ letters: $21$ consonants and $5$ vowels).  

a. How many ways can you construct strings with no vowel?
b. How many ways can you construct strings with at least one vowel?  

This is what I did for both of them... however I think at least one of them is incorrect.
a. Ways with no vowel = $21^3$ since for each letter, we have $21$ options.  
b. Ways with at least one vowel = total ways with no restrictions - ways with no vowel
= $26^3 - 21^3$.  
However, manually computing it:
Ways with at least one vowel = ways with one vowel + ways with two vowels + ways with three vowels 
= $\binom{5}{1} \times \binom{21}{2} \times 3! + \binom{5}{2} \times \binom{21}{1} \times 3! + \binom{5}{3} \times \binom{21}{0} \times 3!$  
which doesn't equal to the other way.

REPLY [3 votes]: In the second case, you are not taking into account repeats. For example your way of calculating the options for making a word with 2 consonants only allows 2 different consonants, not repeats as your definition of a string would indicate.  Calculating the number of cases with one vowel should be $5*21^2*3$
So overall, it should be $5*21^2*3+5^2*21*3+5^3=8315$