Before independence, the Indian subcontinent was divided into British India and various princely states that were under local rulers but subject to British suzerainty.
Step 1: Analyze the options.
(A) 565: This is the officially recognized number of princely states in India at the time of independence. These states were integrated into the Indian Union through the diplomatic efforts of Sardar Vallabhbhai Patel.
(B) 562: While some historical records mention 562 states, the figure of 565 is the standard used in Indian constitutional and historical contexts.
(C) and (D): These figures are historically inaccurate.
Step 2: Conclusion.
The correct number is 565, representing the vast array of kingdoms and estates that existed alongside British provinces.
Final Answer:
\[
\boxed{(A) \, 565}
\]