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.
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.