# How many strings of 8 English letters are there…?

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!

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## 1 Answer

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.

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