String of letters and ways to have at least one vowel 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.

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 $5*21^2*3+5^2*21*3+5^3=8315$