Q:

How many strings of eight English letters are there that contain exactly one vowel, if letters can be repeated?

How many strings of eight English letters are there that contain exactly one vowel, if letters can be repeated?

A:

Out of 5 vowel 1 vowel.
$\binom{5}{1}$
7 letters can out of 21 consonants can be choosen. 
$21^7$
And 1 vowel can be on any 8 places in 8 ways.
Total = $\binom{5}{1} * 21^7 * 8$

