2-Methylcyclohexanone is converted into P by LDA in THF at -78^circC followed by CH3I, and into Q by NaOEt in EtOH at room temperature followed by…

LDA is bulky and strong and, at $-78^\circ\mathrm{C}$, removes the more accessible proton of the less substituted $\alpha$-carbon (kinetic enolate); methylation therefore gives $\mathbf{P}$ = 2,6-dimethylcyclohexanone. $\mathrm{NaOEt}$/$\mathrm{EtOH}$ is reversible and gives the more substituted, more stable (thermodynamic) enolate, so methylation at that carbon gives $\mathbf{Q}$ = 2,2-dimethylcyclohexanone. $\mathbf{R}$ is 2,6-dimethylcyclohexan-1-ol, which has three stereocentres ($\mathrm{C1}$, $\mathrm{C2}$, $\mathrm{C6}$) in a ring with a symmetric substitution pattern. For cis-2,6-dimethyl the $\mathrm{OH}$ may be cis or trans to both methyls: 2 meso compounds. For trans-2,6-dimethyl (chiral) the $\mathrm{OH}$ is necessarily cis to one methyl and trans to the other, and the two choices are the same by the $C_2$ relation, giving one enantiomeric pair: 2 isomers. Total $=2+2=4$. Answer: (C).