I don't understand what you've written. You seem to be putting 3 letters into 2 slots.
Assuming the slots are 1-5 there are 2 even positions, 2 and 4.
There are only 2 ways to put the "vocals letters" (a better English word is vowels) into these slots. I=2, A=4; or A=2, I=4.
Then there are 3! ways of arranging the consonants (the remaining letters).
2*3! = 12
so there are 12 arrangements that have vowels in even numbered slots.
Hello, delso!
The letters 'P, I, H, A, T ' are to be arranged in a row.
Find the number of possible arrangement if the vowels occupy the even positions.
After filling in P, H, T into second and fourth location. . .
HOW did you fit three letters into two positions?
$\displaystyle \text{We have: }\;\_\;\_\;\_\;\_\;\_ $
$\displaystyle \text{Place the two vowels: }\;\_\;\text{v}\;\_\;\text{v}\;\_ $
There are $\displaystyle 2!$ permutations of the vowels.
$\displaystyle \text{Place the three consonants: }\:\text{c}\;\text{v}\;\text{c}\;\text{v}\;\text{c }$
There are $\displaystyle 3!$ permutations of the consonants.
Therefore, there are: .$\displaystyle (2!)(3!) \,=\,12$ possible arrangements.