Find the number of 5-letter “words” (strings with length 5) consisting of 3 different consonants and 2 different vowels. The English alphabet has 26 letters of which 5 are vowels.

Expert Answers
mlehuzzah eNotes educator| Certified Educator

I can only answer one of these questions, so I have edited your answer to reflect that:

If there are no restrictions, then we can describe a word by first describing in which position the vowels occur.  If we think of the word as being made up of vowels, V, and consonants, C, then a word can be respresented by rearranging 2Vs and 3Cs, for example:

VVCCC, or VCVCC, or CCVCV, etc.

There are `5!/(2!3!) = 10` ways to rearrange the letters.

Once we have picked the places in which the vowels/consonants occur, we can do the following:

there are 21 choices for the first C, 21 for the second, and 21 for the third.  There are 5 choices for the first V, and 5 for the second.  Thus we have a total of:

10*21*21*21*5*5=2315250 possible "words"



PS: some hints:

To find the number of words with a B, take the total number of possible words and subtract the ones with no B.  The ones with no B are essentially those from an alphabet with only 25 letters (20 consonants).

To find the number of words with no B or C, essentially think of your alphabet as having 24 letters (19 consonants)

To find the number that start with a B, let your "word" be the last 4 letters.  Then you are basically looking for 4 letter words containing 2 vowels and 2 consonants.