Step 1: Identify the historical timelines of each author.
\begin{itemize}
\item Patanjali (d): 2nd century BCE – Famous for his work on Sanskrit grammar (Mahābhāṣya).
\item Chanakya (e): 4th century BCE – Also known as Kautilya, author of Arthashastra, advisor to Chandragupta Maury(A)
\item Kalidasa (c): 4th–5th century CE – Renowned classical Sanskrit poet and playwright.
\item Kalhana (b): 12th century CE – Historian and author of Rajatarangini.
\item Banabhatta (a): 7th century CE – Court poet of Harshavardhana, known for Harshacharit(A)
\end{itemize}
Step 2: Arrange in chronological order (from earliest to latest):
\[
\text{Patanjali (d)} \rightarrow \text{Chanakya (e)} \rightarrow \text{Kalidasa (c)} \rightarrow \text{Kalhana (b)} \rightarrow \text{Banabhatta (a)}
\]
Hence, the correct order is:
\[
d \rightarrow e \rightarrow c \rightarrow b \rightarrow a
\]