Concept:
Ancient Indian philosophy developed through six orthodox systems known as the Shad Darshanas. Each philosophical school had a founding sage or an important systematizer associated with it.
Step 1: Match the Nyaya School.
The Nyaya School is associated with:
\[
Gautama
\]
Nyaya philosophy mainly emphasizes logic and reasoning.
Thus,
\[
A \rightarrow III
\]
Step 2: Match the Vaisheshika School.
The Vaisheshika School was founded by:
\[
Kanada
\]
This school is famous for atomistic theory.
Hence,
\[
B \rightarrow II
\]
Step 3: Identify the Sankhya School philosopher.
The Sankhya School is associated with:
\[
Ishvar Krishna
\]
He systematized Sankhya philosophy in the work Sankhyakarika.
Therefore,
\[
C \rightarrow IV
\]
Step 4: Match the Vedanta School.
The Vedanta School is associated with:
\[
Badarayana
\]
He composed the Brahma Sutras, which form the basis of Vedanta philosophy.
Thus,
\[
D \rightarrow I
\]
Step 5: Combine all correct pairs.
Hence:
\[
A-III,\; B-II,\; C-IV,\; D-I
\]
Therefore:
\[
\boxed{\text{Option (4)}}
\]
Elimination Method:
• Options (1), (2), and (3) contain incorrect philosophical associations.
• Option (4) perfectly matches all schools and philosophers.