Step 1: Identify the functional groups in the starting compound.
The given compound contains a benzylic secondary alcohol and a terminal primary alcoholThe aromatic ring also contains a methoxy group, but it does not participate directly in the given sequence
Step 2: Action of MnO$_2$.
MnO$_2$ is a selective oxidizing agentIt oxidizes benzylic and allylic alcohols efficiently under mild conditions
Step 3: Formation of ketone.
The benzylic secondary alcohol is oxidized to a ketone, while the terminal primary alcohol remains unchanged under these conditions
\[
\text{Ar-CH(OH)-CH}_2\text{CH}_2\text{OH}
\xrightarrow{\text{MnO}_2}
\text{Ar-CO-CH}_2\text{CH}_2\text{OH}
\]
Step 4: Action of PBr$_3$.
PBr$_3$ converts alcohols into alkyl bromidesTherefore, the terminal primary alcohol is converted into the corresponding bromide
\[
\text{Ar-CO-CH}_2\text{CH}_2\text{OH}
\xrightarrow{\text{PBr}_3}
\text{Ar-CO-CH}_2\text{CH}_2\text{Br}
\]
Step 5: Action of NaOEt in ethanol.
NaOEt acts as a strong baseIt abstracts an acidic $\alpha$-hydrogen from the carbon next to the carbonyl group
Step 6: Elimination reaction.
Removal of the $\alpha$-hydrogen followed by elimination of bromide gives an $\alpha,\beta$-unsaturated ketoneThis product is stabilized by conjugation with the carbonyl group
\[
\text{Ar-CO-CH}_2\text{CH}_2\text{Br}
\xrightarrow{\text{NaOEt}}
\text{Ar-CO-CH=CH}_2
\]
Step 7: Conclusion.
Thus, the final product is the conjugated enone shown in option (A)
\[
\boxed{\text{A}}
\]