TITLE: How many strings of 8 English letters are there...?
QUESTION [6 upvotes]: 1) That contain at least one vowel, if letters can be repeated?
$26^8-21^8$
2) That contain exactly one vowel, if letters can be repeated?
$8\cdot 5\cdot 21^7$
3) That start with an X and contain at least one vowel, if letters can be repeated?
$1\cdot 26^7-1\cdot 21^7$
Assume only upper-cased letters are used.
I'm just trying to intuitively understand what's going on here.  Can anyone explain in a clear and concise manner?
Thank you!

REPLY [11 votes]: There are $26$ letters, of which $21$ are consonants and $5$ are vowels.
1) There are $26^8$ words in all, and $21^8$ of them contain only consonants; all others contain at least one vowel.
2) There are $8$ positions for the vowel, $5$ options for the vowel and $21^7$ options for the $7$ consonants.
3) Same as 1), except one letter is fixed so there are only $7$ left.