SOLUTION: How many ways are there to rearrange the letters in the word aptitude, if the first and last letter must each be a vowel?

Question 390902: How many ways are there to rearrange the letters in the word aptitude, if the first and last letter must each be a vowel?
Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
i'll take a stab at this:
A list of the possible 1st letters with the possible last letters:
a - - - -i, u, or e (3 choices)
i - - - - a, u, or e (3 choices)
u - - - -a, i, or e (3 choices)
e - - - -a, u, or i (3 choices)
This is 3 choices times 4, or 12 choices for the 1st and last
letters being vowels.
Suppose I pick
a - - ptitud - - e for my 1st choice
For the 2nd letter I have 6 choices
For the 3rd letter I have 5 choices remaining
For the 4th letter I have 4 choices remaining
For the 5th letter I have 3 choices remaining
For the 6th letter I have 2 choices remaining
For the 7th letter I have 1 choice remaining
Since the letters are 1st AND 2nd AND 3rd . . .etc.
I must multiply the choices

For each of my choices I have treated t and t as different
letters, but they are not different, so each arrangement is
counted twice, like:
a - - tuitdp - - e and
a - - tuitdp - - e
So if I divide by , I'll get the actual number of
arrangements I want

Each of these can be matched with 1 of the 12
ways I can pick letters for 1st and last, so
is my answer
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