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 two-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 QO, XY, and UZ.)
We can use any vowel = 6 choices and any consonant = 20 choices
And the vowel can come first or the consonant can come first = 2 choices
So.....with one vowel we have 6 * 20 * 2 = 240 "words"
We can also use a word with the same vowel repeated = 6 "words"
We can also select any one of the vowels first = 6 choices and any one of the remaining vowels second = 6 * 5 = 30 "words"
So...... 240 + 6 + 30 = 276 "words"