Step 1: Understanding the Concept:
Chronological order requires identifying the years of establishment for various medieval Indian dynasties and kingdoms.
Step 2: Detailed Explanation:
Let's list the establishment dates for each:
1. E. Establishment of the Vijayanagara Empire: Founded by Harihara I and Bukka Raya I in 1336 CE.
2. B. Establishment of the Bahmani kingdom: Founded by Ala-ud-Din Bahman Shah in 1347 CE.
3. D. Establishment of the Gajapati kingdom of Orissa: Founded by Kapilendra Deva in 1434 CE.
4. C. Emergence of the Sultanates of Ahmadnagar: One of the five Deccan Sultanates that emerged after the fragmentation of the Bahmani kingdom, established around 1490 CE by Malik Ahmad.
5. A. Establishment of the Mughal empire by Babur: Established after the First Battle of Panipat in 1526 CE.
The chronological sequence is: 1336 (E) $\rightarrow$ 1347 (B) $\rightarrow$ 1434 (D) $\rightarrow$ 1490 (C) $\rightarrow$ 1526 (A).
Step 3: Final Answer:
The correct order is E, B, D, C, A.