The reaction proceeds as follows:
Step 1: The triple bond in CH$_3$C$\equiv$C-CH$_3$ is reduced to a trans-alkene using Na/liq. NH$_3$:
\[\text{CH}_3\text{C} \equiv \text{C-CH}_3 \xrightarrow{\text{Na/liq. NH}_3} \text{CH}_3\text{CH} = \text{CHCH}_3\]
Step 2: The trans-alkene undergoes oxidation with dilute KMnO$_4$ at 273 K. This leads to the formation of a vicinal diol:
\[\text{CH}_3\text{CH} = \text{CHCH}_3 \xrightarrow{\text{dil. KMnO}_4} \text{CH}_3\text{CH(OH)CH(OH)CH}_3\]
The final product `P' contains two hydroxyl (-OH) groups, contributing two oxygen atoms.
Number of oxygen atoms in product `P':
\[2 \, (\text{from hydroxyl groups})\]