Concept:
Indian literary traditions evolved over a long historical period beginning from the Vedic age and gradually developing into philosophical, mythological, poetic, and dramatic literature.
To arrange them chronologically, we must identify the approximate historical sequence of their development.
Step 1: Identify the earliest literature.
\[
\text{Rigveda}
\]
is the oldest Vedic literature and one of the earliest known texts of India.
Thus, the first position is:
\[
B
\]
Step 2: Identify the philosophical literature.
The:
\[
\text{Upanishadas}
\]
developed later as philosophical interpretations of Vedic thought.
Thus, the next position is:
\[
A
\]
Step 3: Identify the mythological literature.
The:
\[
\text{Puranas}
\]
were composed later and contain mythology, legends, and religious narratives.
Thus, the next position is:
\[
D
\]
Step 4: Identify the poetic and dramatic literature.
\[
\text{Kavya}
\]
refers to classical poetic literature which flourished in the later classical period.
\[
\text{Natya}
\]
or dramatic literature developed as a specialized literary and performance tradition.
Thus, the sequence continues as:
\[
C \rightarrow E
\]
Step 5: Arrange the complete chronological order.
The correct sequence becomes:
\[
B \rightarrow A \rightarrow D \rightarrow C \rightarrow E
\]
Step 6: Compare with the given options.
Among the given choices:
\[
(A)\ A,B,C,D,E
\]
\[
(B)\ E,B,D,C,A
\]
\[
(C)\ B,A,D,C,E
\]
\[
(D)\ A,B,E,C,D
\]
the correct sequence is:
\[
\boxed{B,A,D,C,E}
\]
Hence:
\[
\boxed{\mathrm{(C)}}
\]