Out of the 8 possible positions, we choose 3 in order to place vowels U, I, E, and we have C(3,3) ways of choosing the vowels, i.e. only one way - their required order. Then we arrange the remaining in 5!/2! (we divide by 2! because of the repetition of letter L) and then we subtract 1, as required by the question.

That is,

C*(8,3)*C(3,3)*5!/2! -1 = 3359.