Question

The English alphabet contains 21 consonants and fivevowels. How many strings of six lowercase letters of theEnglish alphabet contain:

a) exactly two vowels?

b) at least two vowels?

Answer 1

a)

The location of the vowels can be chosen in $\dbinom{6}{2}$ ways.

For each choice of locations, the locations can be filled with vowels in $5^2$   ways.

Four empty locations can be filled with consonants in $21^4$ ways.

$$\dbinom{6}{2}(5)^2(21)^4=72930375$$

b)

The total number of strings is $26^6=308915776$

The number of strings which contain exactly vowels is $21^6=85766121$

Exactly 1vowel: The location of the vowel can be chosen in $\dbinom{6}{1}$ ways.

For each choice of location, the location can be filled with a vowel in $5^1$ ways.

That leaves $5$ empty locations, which can be filled with consonants in $21^5$ways

$$\dbinom{6}{1}(5)^1(21)^5=122523030$$

The number of strings which contain at least two vowels is

$$308915776-85766121-122523030$$

$$=100626625$$