Step 1: Understand the question:
The question asks us to pair four ancient Indian branches of knowledge with the subject each one deals with.
We first match each item on its own, then look for the option that agrees with all four pairs.
Step 2: Match each item from what it covers:
Anvikshiki is the old science of reasoning and inquiry, so it goes with Logic (III).
Dharmashastra is the body of texts on duty, law and social conduct, so it goes with Sociology (II).
Kamashastra is the tradition on love, desire and pleasure, so it goes with Erotics (I).
Arthashastra is the tradition on statecraft, wealth and governance, so it goes with Polity (IV).
The full match is A-III, B-II, C-I, D-IV.
Step 3: Check Option 1:
Option 1 says A-I. But Anvikshiki is about logic, not erotics. This option is wrong.
Step 4: Check Option 2:
Option 2 says A-III, B-II, C-I, D-IV. Every pair agrees with our match. This option is correct.
Step 5: Check Option 3:
Option 3 says A-I and C-IV. Both pairs are wrong, because Kamashastra is not about polity. This option is wrong.
Step 6: Check Option 4:
Option 4 says A-III, which is right, but B-IV puts Dharmashastra with Polity, and D-II puts Arthashastra with Sociology. Both are swapped. This option is wrong.
Final Answer:
The only option that matches all four pairs is Option 2. \[ \boxed{A\text{-III, B-II, C-I, D-IV (Option 2)}} \]