Here are the reasons for the correct statements:

(A) The Ain is made up of five books: The 'Ain-i-Akbari' is indeed composed of five books. It is a Persian text written by Abu'l-Fazl ibn Mubarak, a courtier in the Mughal Emperor Akbar's court. The five books cover various aspects of Akbar's empire, including the administration, culture, and society.

(B) Ain was officially sponsored to record detailed information: 'Ain-i-Akbari' was officially sponsored and commissioned by Emperor Akbar. It was intended to be a comprehensive record of the Mughal Empire, its administration, and various aspects of the society and culture.

(D) Historians have detected numerous errors in totaling, which are generally minor: While 'Ain-i-Akbari' is a valuable historical document, historians have identified some errors in the numerical data and calculations presented in the text. These errors are typically considered minor and do not significantly detract from the overall historical value of the work.