Concept:
This question tests the chronology of important Indian historical and artistic sites from prehistoric cave paintings to early medieval temple centres.
Step 1:
Bhimbethka belongs to the prehistoric period and contains Paleolithic rock shelters and cave paintings.
It is the earliest site in the sequence.
Step 2:
Ajanta caves developed mainly between the 2nd century BCE and 6th century CE.
Known for Buddhist paintings and rock-cut caves.
Step 3:
Bagh caves belong mainly to the Gupta/post-Gupta period around the 5th--6th century CE.
Step 4:
Badami cave temples were developed under the Chalukyas during the 6th--7th century CE.
Step 5:
Kanchipuram became a major Pallava temple centre mainly during the 7th--8th century CE.
Thus the correct chronology is:
\[
A \rightarrow E \rightarrow B \rightarrow D \rightarrow C
\]
Hence, the correct answer is:
\[
\boxed{\text{(1) A, E, B, D, C}}
\]