The number of ways in which the letters of the word MULTIPLE be arranged without changing the order of the vowels is

Answer 1

There are 3 vowels, so the 5 consonants can be divided into groups of
0,0,0,5
0,0,1,4
0,0,2,3
...
5,0,0,0
For each of those groups i,j,k,l the consonants can be permuted in
i! * j! * k! * l! ways
Then divide by 2, since the 2 L's are indistinguishable.

Maybe someone else can think of a simpler way.