Step 1: Understand the question.
Among the given amines, we must pick the strongest base in water. A stronger base donates its lone pair more readily.
Step 2: Sort the amines by type.
$\text{CH}_3\text{NH}_2$ is primary ($1^\circ$), $(\text{CH}_3)_2\text{NH}$ is secondary ($2^\circ$), and both $(\text{CH}_3)_3\text{N}$ and $(\text{C}_2\text{H}_5)\text{N}(\text{CH}_3)_2$ are tertiary ($3^\circ$).
Step 3: List the three factors that fight each other.
Basicity in water depends on (a) the push of electrons from alkyl groups, which helps, (b) how well the protonated ion is wrapped by water (solvation), which helps, and (c) crowding of bulky groups, which blocks the proton and hurts.
Step 4: Apply these to methyl amines.
A primary amine is well solvated but has weak electron push. A tertiary amine has strong push but is crowded and poorly solvated. A secondary amine sits in the sweet spot: good push plus decent solvation.
Step 5: Write the known order.
For methyl-substituted amines in water the order is \[ 2^\circ > 1^\circ > 3^\circ > \text{NH}_3 \]
Step 6: Pick the strongest.
The only secondary amine here is $(\text{CH}_3)_2\text{NH}$, so it is the strongest base, option (C).
\[ \boxed{(\text{CH}_3)_2\text{NH}} \]