Combinations with words in the alphabet

Question

Suppose that a “word” is any string of seven letters of the alphabet, with repeated letters allowed. How many words have exactly one vowel?

I'm having some trouble trying to get the right steps:

Ways to choose one vowel $C_{(1,5)} = 5,$
$7$ positions, $1$ being used: $5\cdot 21^6.$

Are these the right steps so far?

Answer 1

The thinking steps can be:

$$(\textrm{Choose a vowel})\cdot(\textrm{Choose a position for the vowel})\cdot(\textrm{other works}),$$

so your first step is right, there are $C_{(1,5)}=5$ ways to choose one vowel. But you didn't choose a position for it, i.e.

$$\binom{7}{1}=7,$$

finally there are six remaining positions for consonants, and you did right that is $21^6$.

Answer 2

There are 7 ways to place the vowel letter.

$5\times7\times 21^6$