Nalanda was one of the most famous ancient universities of India.
It was a major Buddhist center of learning.
So statement A is correct.
Chanakya is generally associated with Takshashila, not Nalanda.
So statement B is incorrect.
Takshashila was an ancient center of learning where subjects such as politics, warfare, medicine, grammar and philosophy were taught.
It is also associated with Brahminical learning.
So statement C is correct.
Nalanda was located in present-day Bihar, not on the banks of the Narmada.
So statement D is incorrect.
Takshashila was located in present-day Pakistan, not Nepal.
So statement E is incorrect.
Therefore, the correct statements are:
\[
A\ \text{and}\ C\ \text{only}.
\]