Before the independence and partition of India in 1947, the Indian subcontinent consisted of British India (provinces directly ruled by the British) and a vast number of semi-autonomous states ruled by local monarchs.
Step 1: Analyze the options.
(A) 565: This is the widely accepted historical figure used by the Government of India and the Ministry of States. These states covered approximately 40% of the area of pre-independence India and were integrated into the Indian Union primarily through the efforts of Sardar Vallabhbhai Patel.
(B) 562: Some historical sources cite 562 as the number of states that were officially recognized as part of the Chamber of Princes, but 565 is the standard educational and governmental figure.
(C) 620: This figure is significantly higher than the recorded number of recognized political entities under the British Raj.
(D) 728: This number is incorrect and does not align with historical data regarding the integration of India.
Step 2: Conclusion.
There were 565 princely states in India before independence, ranging from large states like Hyderabad to very small estates.
Final Answer:
\[
\boxed{(A) \, 565}
\]