
Reaction sequence analysis:
Step 1: Ethylbenzene oxidation: \(\text{C}_6\text{H}_5\text{CH}_2\text{CH}_3\) reacts with \(\text{KMnO}_4\) and \(\text{KOH}\) under heating to form potassium benzoate, \(\text{C}_6\text{H}_5\text{COOK}\) (A).
\[\text{C}_6\text{H}_5\text{CH}_2\text{CH}_3 \rightarrow [\text{KMnO}_4, \text{KOH}, \Delta] \text{C}_6\text{H}_5\text{COOK} (A).\]
Step 2: Acidification: The carboxylate salt \(\text{C}_6\text{H}_5\text{COOK}\) (A) is treated with \(\text{H}_3\text{O}^+\) to yield benzoic acid, \(\text{C}_6\text{H}_5\text{COOH}\) (B).
\[\text{C}_6\text{H}_5\text{COOK} + \text{H}_3\text{O}^+ \rightarrow \text{C}_6\text{H}_5\text{COOH} (B).\]
Step 3: Bromination: Benzoic acid (B) undergoes electrophilic aromatic substitution with \(\text{Br}_2\) and \(\text{FeBr}_3\), resulting in para-bromobenzoic acid, \(p-\text{BrC}_6\text{H}_4\text{COOH}\) (C), due to the directing effect of the carboxylic acid group.
\[\text{C}_6\text{H}_5\text{COOH} \rightarrow [\text{Br}_2, \text{FeBr}_3] p-\text{BrC}_6\text{H}_4\text{COOH} (C).\]
Step 4: \(\pi\)-bond count in C: The molecule \(p-\text{BrC}_6\text{H}_4\text{COOH}\) (C) contains 3 \(\pi\)-bonds in the benzene ring and 1 \(\pi\)-bond in the carbonyl group (\(\text{C} = \text{O}\)).
Total \(\pi\)-bonds = 3 (from ring) + 1 (from \(\text{C} = \text{O}\)) = 4.
Final Answer: 4.

