MHB Strings of eight English letters

[alexmahone]

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

My attempt:

Let us first find the number of strings that contain at least one 'a' and no other vowels.
Total number of strings including 'a' but excluding the other vowels $=22^8$
Number of strings without any vowels $=21^8$
So, number of strings that contain at least one 'a' and no other vowels $=22^8-21^8$

Sincere there are five vowels, we get $5(22^8-21^8)=85,265,070,875$. The book's answer is $72,043,541,640$. Where did I go wrong?

 

[MarkFL]

This is how I would approach the problem...

Suppose the vowel is first, so we have 5 choices for the first letter, and the 21 letters for each of the remaining 7 letters, and applying the counting rule, we find:

$$N_1=5\cdot21^7$$

But now, we consider that the vowel may be anywhere, in 1 of 8 places, hence:

$$N=8\cdot N_1=72043541640$$

 

[alexmahone]

This is how I would approach the problem...

Suppose the vowel is first, so we have 5 choices for the first letter, and the 21 letters for each of the remaining 7 letters, and applying the counting rule, we find:

$$N_1=5\cdot21^7$$

But now, we consider that the vowel may be anywhere, in 1 of 8 places, hence:

$$N=8\cdot N_1=72043541640$$

Thanks.

But can't the vowel occur more than once? After all the question says "if letters can be repeated".

 

[MarkFL]

Thanks.

But can't the vowel occur more than once? After all the question says "if letters can be repeated".

The problem states:

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

I took this to mean that only one of the eight letters can be a vowel. :)