How many times the word DIRECTOR can be rearrange without changing the order of vowels?
There are 2 ways of solving this problem.

The problem says we shouldn’t change the order of the vowels, the vowels in the question are IEO and we have to maintain the same order. So, 3 characters cannot be rearranged. The rest can still be rearranged. So Let’s consider 8 empty boxes where we need to fill the 8 characters we have.
We have the consonants DRCTR. Let us first arrange that. The number of ways to do that is first you choose 5 spaces out of 8 (which can be done in 8C5 ways), and fill those 5 spaces with the consonants ( filling 5 boxes with 5 consonants where two of them are similar  2 Rs  5P5 / 2! ).
The above might be one of the possibilities of what we just calculated so far. 5 boxes were selected and filled with consonants. Now, of the 3 boxes, we don’t have a choice to arrange the vowels. So the order will be IEO which is only 1 possible way. So, your final answer will be (8C5 x 5P5) / (2!) 
Second way is to assume that all your vowels are the same (meaning that there is only one way of arranging them). So your word has 8 characters and 2 of them are of the same kind (Rs) and 3 of them are of the same kind (vowels) as well. So your final answer will be (8P8) / (2! * 3!)