In how many permutations of word "LOGARITHM" vowels have alphabetical order? The answer is $9 \cdot 8 \cdot 7$ but I can't process it .
Presuming you mean just that the A has to be before the I and the I before the O in the permutation, I don't think the answer is 9*8*7.
Think about it this way: for every "good" permutation, how many "bad" ones are there where the vowels are in the same positions but our of order? For instance the good permutation ALGIROTHM has a matching bad permutation ALGORITHM.
Now if you know how many permutations there are altogether, and how many bad permutations there are for every good one, you can get the answer.