English has 6 letters that can be vowels. This includes Y, which can be either a consonant or a vowel; for this problem, we'll consider Y a vowel.
The other 20 English letters are always consonants.
How many three-letter "words" can we make from these letters if we are required to use at least one vowel? (We aren't limited to words that have an actual meaning in English. Thus, for this problem, we'll include nonsense "words" like QOP, XYZ, and UTZ.)
There are 26 letters from which you can make: 26 nPr 3 =15,600 three-letter "words". Each one of these "words" begins with one of the 26 letters this many times: 15,600 / 26 =600 three-letter beginning with each letter of the alphabet. Since you have "6" vowels, then you will have: 600 x 6 =3,600 three-letter "words" with at least one vowel.
If repeats are allowed, then you will have: 26^3 =17,576 three-letter "words". Same as above:17,576 / 26 =676 "words" beginning with each letter. And 676 x 6 vowels =4,056 "words" with at least 1 vowel.
