Step 1: Understanding the Concept:
This problem involves identifying a starting material through a series of nitrogen-based organic reactions, including amide formation, the Hofmann Bromamide degradation, and the Carbylamine reaction.
Step 2: Detailed Explanation:
Working backward from the final product, Methyl Isocyanide ($CH_3-NC$):
1. The Carbylamine reaction (treatment with $CHCl_3$ and $KOH$) produces an isocyanide from a primary amine. Therefore, the precursor to $CH_3-NC$ must be Methylamine ($CH_3-NH_2$).
2. The Hofmann Bromamide degradation (treatment with $Br_2$ and $KOH$) produces a primary amine with one fewer carbon atom from an amide. To get $CH_3-NH_2$, the precursor amide 'Q' must be Ethanamide ($CH_3CONH_2$).
3. 'Q' ($CH_3CONH_2$) is formed by reacting 'P' with $NH_3$ and heating. This is a standard synthesis of amides from carboxylic acids. Therefore, 'P' must be Ethanoic acid (Acetic acid), $CH_3COOH$.
Step 3: Final Answer:
Compound 'P' is CH₃-C(O)-OH.