Arranging letters of the word PROPORTIONALITY so that vowels and consonants still occupy the same places? In how many ways can the letters of the word PROPORTIONALITY be arranged so that the vowels and consonants still occupy the same places? Can someone help me understand what this question even means in terms of,so for example does it mean the specific vowels have to be in the identical space or any vowel in general has to occupy the same space.

It seems like the question means that the vowels/consonants can move freely among themselves, so long as they don't go in each others spaces. In this case you can decompose the problem into two smaller problems, just thinking about the vowels and the consonants.

There are 6 vowels, with O repeated 3 times and I repeated 2 times. Therefore there are $\frac{6!}{3!2!}$ ways of arranging them amongst themselves, since O's and I's are indistinguishable.

Similarly there are $\frac{9!}{2!2!2!}$ ways of arranging the consonants, since the R, P, and T appear twice.

Multiplying these together gives a final answer of $\frac{6!9!}{(2!)^{4}3!}$