Goal: Convert Ethyl Bromide ($C_2H_5Br$) to Propanoic Acid ($C_2H_5COOH$).
This involves ascent of series (adding one Carbon).
Method II (Nitrile Synthesis):
$C_2H_5Br \xrightarrow{KCN} C_2H_5CN \xrightarrow{H_3O^+} C_2H_5COOH$.
This is a standard valid method.
Method III (Grignard Synthesis):
$C_2H_5Br \xrightarrow{Mg, Ether} C_2H_5MgBr \xrightarrow{CO_2} C_2H_5COOMgBr \xrightarrow{H_3O^+} C_2H_5COOH$.
This is also a standard valid method to increase carbon chain by one and form carboxylic acid.
Method I (Oxidation):
$C_2H_5Br \xrightarrow{OH^-} C_2H_5OH \xrightarrow{KMnO_4} CH_3COOH$ (Ethanoic acid).
This does not increase the carbon chain length. It yields a 2-carbon acid, not propanoic (3-carbon). Incorrect.
Conclusion:
Methods II and III work.