Combination Problem Involving Alphabet

The English alphabet contains 21 consonants and 5 vowels. How many strings of six lowercase letters of the alphabet contain:

Exactly one vowel?

Exactly two vowels?

At least one vowel?

At least two vowels?

I tried using C(26,6); though I know this will only give me the number of 6-combinations with 26 elements.

Letter strings: vowels and consonants

Hi -

The key fact here is that letters and vowels may be repeated. Each of the six spaces in the line may be filled in 26 ways, if we're not concerned about vowels and consonants at this stage. So the total number of different strings altogether is $\displaystyle 26^6$.

In parts one and two, you need to:

- (a) choose which space(s) will be occupied by vowels;
- (b) choose which vowel(s) to put in these spaces;
- (c) fill the remaining empty spaces with consonants.

So work out how many ways there are of carrying out tasks (a) to (c) in each case, and multiply them together.

In part three, subtract the number of letter strings containing *all consonants* from the total number of strings altogether. This will give you how many contain at least one vowel.

In part four, subtract from your answer to part three the number of strings containing *exactly *one vowel (i.e. your answer to part one). This will give the number of strings containing at least two vowels.

Can you see how to do it now?

Grandad