Step 1: Understanding behavior of metals with conc. HNO\(_3\).
Concentrated nitric acid is a strong oxidizing agent and generally dissolves most metals, but some metals become passivated or show resistance.
Step 2: Checking each metal.
- Cu: dissolves in HNO\(_3\) → forms Cu\(^{2+}\), so it reacts
- Al: becomes passivated due to protective Al\(_2\)O\(_3\) layer → does not dissolve easily
- Zn: reacts with HNO\(_3\) → dissolves
- Ni: shows passivation in conc. HNO\(_3\) → does not dissolve readily
- Fe: also becomes passivated → does not dissolve
Step 3: Identifying non-dissolving metals.
Metals that do not dissolve: Al, Ni, Fe → total = 3
Step 4: Final reasoning.
Passivation occurs due to formation of protective oxide layers preventing further reaction with acid.
Step 5: Final conclusion.
Hence, number of metals not dissolving = 3.
Final Answer:
\[
\boxed{3}
\]