Step 1: Sort the list by place.
Four places are on offer: New Delhi, Chennai, Mumbai and the Telugu states. Each institution must go to exactly one.
Step 2: Start with the easy ones.
IGNCA is a national body, so it sits in the capital, New Delhi. Prithvi Theatre is the famous playhouse of Mumbai.
Step 3: Fill the remaining two.
Koothu-p-Pattrai is a Tamil theatre group, so Chennai fits it. Surabhi is the touring Telugu company, so Andhra Pradesh and Telangana is left for it.
Step 4: Check against the options.
Only option (1) has A-III, B-I, C-IV and D-II together. Options (2) and (3) put Koothu-p-Pattrai in New Delhi, and option (4) puts it in the Telugu states and Surabhi in New Delhi. Each of them has an error.
Step 5: Conclude.
Option (1) is the answer.
Final Answer:
Option (1) is correct.
\[ \boxed{\text{1}} \]