The task requires arranging historical events in chronological order. Let's break down the mentioned events and list their likely chronological order for clarity:
- (D) Rising in Lucknow: This refers to the onset of the rebellion in Lucknow, a critical phase of the 1857 uprising.
- (B) British suffer defeat in the Battle of Chinhat: This event occurred during the 1857 conflict when the British faced a setback.
- (E) British forces under Havelock and Outram enter the Residency in Lucknow: This highlights the British effort to relieve and reinforce their position in Lucknow.
- (C) Shah Mal killed in battle: Given the sequence, Shah Mal's death in battle appears to follow the intervention in Lucknow.
- (A) Rani Jhansi killed in battle: Known for her role in the 1857 uprising, the death of Rani Jhansi follows later in the sequence.
Based on the analysis, the correct chronological order of events is (D), (B), (E), (C), (A), which aligns with the provided option. This sequence reflects the progression of key events during the 1857 Indian rebellion.