Step 1: Identify the type of reaction.
Sodium iodide is dissolved in dry acetone and mixed with methyl chloride. This is a halogen exchange reaction, commonly called the Finkelstein reaction, used to swap a chlorine or bromine for an iodine.
Step 2: Balance the equation by atom count.
On the left side we have $\text{CH}_3\text{Cl}$ and $\text{NaI}$. Sodium must pair with the leaving chloride to give $\text{NaCl}$, and the methyl group must pair with iodide to give $\text{CH}_3\text{I}$. \[ \text{CH}_3\text{Cl} + \text{NaI} \rightarrow \text{CH}_3\text{I} + \text{NaCl} \]
Step 3: Explain why the reaction goes to completion.
Dry acetone dissolves $\text{NaI}$ well but barely dissolves $\text{NaCl}$. As $\text{NaCl}$ forms it drops out of solution as a solid, so there is nothing to pull the products back to reactants.
Step 4: Match to the given options.
Only $\text{A} = \text{CH}_3\text{I}$ and $\text{B} = \text{NaCl}$ matches this atom balance and mechanism; the other options introduce atoms that are not present anywhere in the reactants.
\[ \boxed{\text{A} = \text{CH}_3\text{I}, \ \text{B} = \text{NaCl}} \]