Not sure what this means, but I'll take a guess . . .Consider the letters of the word: HEXAGON
a) How many ways can the vowels of the word be arranged using all the vowels?
I assume the consonants stay in place, and we are to permute the vowels.
Then we have: .H _ X _ G _ N
Then the vowels can be placed in: . ways.
b) How many arrangements of the word begin with a vowel?
There are 7 letters: 3 vowels and 4 consonants.
There are 3 choices of vowels for the first letter.
. . Then the other 6 letters can be arranged in ways.
Therefore, there are: . arrangements.
c) How many 3-letter "words" can be made if every "word" must have the pattern
consonant-vowel-consonant and if the letters can be repeated?
For the first letter, there are 4 choices for the consonant.
For the second letter, there are 3 choices for the vowel.
For the third letter, there are 4 choices for the consonant.
Therefore, there are: . possible "words".