Combinatorics question about english letters (with consonants and vowels) The english alphabet contains $21$ consonants and $5$ vowels. How many strings of $6$ lowercase letters of the English alphabet contain 


*

*a) exactly 1 vowel

*b) exactly 2 vowels

*c) at least one vowel

*d) at least two vowels

 A: Exactly $1$ vowel: The location of the vowel can be chosen in $\binom{6}{1}$ ways. (Of course this is $6$, but we are trying to use a technique that works more generally.)
For each choice of location, the location can be filled with a vowel in $5$ ways. 
That leaves $5$ empty locations, which can be filled with consonants in $21^5$ ways. That gives a total of 
$$\binom{6}{1}(5)(21^5).$$
Exactly $2$ vowels: The location of the vowels can be chosen in $\binom{6}{2}$ ways.  
For each choice of locations, the locations can be filled with  vowels in $5^2$ ways. 
That leaves $4$ empty locations, which can be filled with consonants in $21^4$ ways. That gives a total of 
$$\binom{6}{2}(5^2)(21^4).$$
At least $1$ vowel: There are $26^6$ $6$-letter "words." And there are $21^6$ all consonant words. So there are $26^6-21^6$ words that have at least one vowel. 
There are other ways of counting this, but they are less efficient.  
At least $2$ vowels: Again, there are various ways of counting. An efficient way is to count the words that have $0$ vowels or $1$ vowels, and subtract from the total number of words. This approach lets us recycle previous results, and recycling is a virtue.
So take the total number $26^6$ of words, and subtract the $21^6$ all consonant words, and the number of $1$-vowel words we already calculated. 
