Step 1: Understanding the Concept:
The carbylamine (isocyanide) test works only for primary amines. When heated with chloroform and alcoholic KOH, \(\text{R-NH}_2\) gives the foul smelling isocyanide \(\text{R-NC}\).
Step 2: Key Formula or Approach:
Classify each amine: methanamine, ethanamine and propan-2-amine are primary. \(N,N\)-Diethylethanamine, \((\text{C}_2\text{H}_5)_3\text{N}\), is tertiary.
Step 3: Detailed Explanation:
The reaction needs two N-H hydrogens to be removed with the dichlorocarbene formed from \(\text{CHCl}_3\) and KOH.
\[ \text{R-NH}_2 + \text{CHCl}_3 + 3\text{KOH} \to \text{R-NC} + 3\text{KCl} + 3\text{H}_2\text{O} \]
A tertiary amine has no N-H, so it cannot form an isocyanide.
Final Answer:
\(N,N\)-Diethylethanamine does not form an isocyanide, option (C).
\[ \boxed{N,N\text{-Diethylethanamine}} \]