Math Help - fundamental counting principal

1. fundamental counting principal

Could someone please help me, i left my homeowrk to the last minute and it due tomorrow. I don't understand how to complete the question? Even if you could just help with some of the question please. Im in applied math 30, and its the probability unit, about the fundamental counting principle.

QUESTION
Consider the lettersof the word HEXAGON
a) How many ways can the vowels of the word be arranged using all the vowels?
b) How many arrangements of the word begain with a vowel?
c) How many 3-letter "words" can be made if every "word" must have the pattern constant-vowel-constant and if the letters can be repeated?

2. Hello, horses7!

Consider the letters of the word: HEXAGON

a) How many ways can the vowels of the word be arranged using all the vowels?
Not sure what this means, but I'll take a guess . . .

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: . $3! \,= \,6$ 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 $6! = 720$ ways.

Therefore, there are: . $3 \times 720 \:=\:2160$ 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: . $4 \times 3 \times 4 \:=\:48$ possible "words".