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.
Best Answer
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 $5*21^2*3+5^2*21*3+5^3=8315$