Concept:
To arrange ancient cave sites chronologically, we compare their broad historical periods.
Early Buddhist caves such as Ajanta Caves 9 and 10 and Pitalkhora are earlier than later Hindu cave temples such as Badami and Ellora.
Step 1: Identify the earliest caves.
Ajanta Caves 9 and 10 are early Buddhist chaitya caves.
They belong to an early phase of Indian rock-cut architecture.
So, the first is:
\[
A = \text{Cave numbers 9 and 10 at Ajanta}
\]
Step 2: Place Pitalkhora after Ajanta.
Pitalkhora is also an early Buddhist cave site.
It is placed after Ajanta Caves 9 and 10 in the given chronological sequence.
So, the second is:
\[
C = \text{Cave number 4 at Pitalkhora}
\]
Step 3: Place Kanheri Caves.
Kanheri Caves developed over a long period and are later than the earliest Buddhist cave phase.
So, the third is:
\[
E = \text{Kanheri Caves}
\]
Step 4: Place Badami Cave 3.
Badami Cave 3 belongs to the Chalukyan period.
It is later than the early Buddhist cave tradition.
So, the fourth is:
\[
D = \text{Cave 3 at Badami}
\]
Step 5: Place Ellora Cave 16.
Ellora Cave 16, also known as the Kailash temple, belongs to a later phase of rock-cut architecture.
So, it comes last:
\[
B = \text{Cave number 16 at Ellora}
\]
Therefore, the correct chronological order is:
\[
A \rightarrow C \rightarrow E \rightarrow D \rightarrow B
\]
Hence:
\[
\boxed{\text{(C) A, C, E, D, B}}
\]