The reason for the chronological order of these great chronicles in the Mughal Empire is based on the reigns of the respective Mughal emperors and the authors who wrote these chronicles:

Babur Nama (B) was written by Babur, the founder of the Mughal Empire, during his own lifetime. It covers the period of his life and his conquests in the early 16th century.

Akbar Nama (A) was written by Abu'l-Fazl ibn Mubarak during the reign of Akbar, who was the third Mughal Emperor. Abu'l-Fazl served as a historian and chronicler in Akbar's court and wrote about Akbar's reign and accomplishments.

Jahangir Nama (E) was written by Emperor Jahangir himself. Jahangir succeeded Akbar and wrote his own chronicle, which gives insights into his own reign during the early 17th century.

Badshah Nama (C) was written by Emperor Shah Jahan, the fifth Mughal Emperor, and covers his reign and accomplishments. Shah Jahan is best known for building the Taj Mahal.

Alamgir Nama (D) was written by Emperor Aurangzeb, the sixth Mughal Emperor, and is a chronicle of his reign. Aurangzeb was one of the last powerful Mughal emperors.

So, the order of these chronicles follows the sequence of Mughal emperors and the authors who wrote about their respective reigns.

So, the correct option is [2].