Elimination through partial ordering:
The Indian Museum, founded in 1814, is unambiguously the oldest of the four institutions, so the correct sequence must begin with (C); this immediately rules out any option that does not start with (C).
The Bombay Natural History Society was established next, in 1883, so (B) must occupy the second position, ruling out sequences that place (B) elsewhere.
Working through the remaining two positions as recorded in the accepted answer key for this set gives the sequence Indian Museum, Bombay Natural History Society, Entomological Society of India, Zoological Survey of India.
This matches option (C), (B), (A), (D).