Step 1: Idea:
We will solve this by the method of elimination.
We pick the pairs we are surest about and use them to remove options one by one.
Step 2: Use the surest pair first:
Arthashastra is the famous work of Kautilya on governance and the state. It belongs with Polity, so D matches IV.
Option 4 pairs D with II, and Option 3 pairs D with III. Both disagree, so Options 3 and 4 are out.
Step 3: Split the two remaining options:
Options 1 and 2 are the ones left. They differ in A and C.
Option 1 says A-I, C-III. Option 2 says A-III, C-I.
Step 4: Decide using the meaning of the words:
Anvikshiki comes from a word meaning inquiry by following the evidence. That is the study of logic.
Kamashastra is the science of kama, which means desire. That is erotics.
So A goes with III and C goes with I, which is Option 2.
Step 5: Confirm B:
Dharmashastra deals with rules for family and society, which is the content of Sociology (II). This is the same in Option 2, so nothing disagrees.
Final Answer:
Elimination leaves Option 2 as the only consistent choice. \[ \boxed{A\text{-III, B-II, C-I, D-IV}} \]