1. We need the number of 3-digit numbers divisible by either $3$ or $4$, but not divisible by $48$.

Let

• $A =$ set of 3-digit numbers divisible by $3$
• $B =$ set of 3-digit numbers divisible by $4$

We need $|A \cup B| - |\text{multiples of }48|$ because every multiple of $48$ is divisible by $3$ and $4$.

2. Count 3-digit multiples of $3$.

The 3-digit numbers run from $100$ to $999$.

Number of multiples of $3$ in this range: $\left\lfloor \frac{999}{3} \right\rfloor - \left\lfloor \frac{99}{3} \right\rfloor = 333 - 33 = 300$

So, $|A| = 300$

3. Count 3-digit multiples of $4$.

$\left\lfloor \frac{999}{4} \right\rfloor - \left\lfloor \frac{99}{4} \right\rfloor = 249 - 24 = 225$

So, $|B| = 225$

4. Count numbers divisible by both $3$ and $4$.

A number divisible by both $3$ and $4$ must be divisible by $\operatorname{lcm}(3,4)=12$

Thus, $|A \cap B| = \left\lfloor \frac{999}{12} \right\rfloor - \left\lfloor \frac{99}{12} \right\rfloor = 83 - 8 = 75$

5. Use inclusion-exclusion to count numbers divisible by $3$ or $4$.

$|A \cup B| = |A| + |B| - |A \cap B|$ $= 300 + 225 - 75 = 450$

6. Count 3-digit multiples of $48$.

$\left\lfloor \frac{999}{48} \right\rfloor - \left\lfloor \frac{99}{48} \right\rfloor = 20 - 2 = 18$

So there are $18$ three-digit numbers divisible by $48$.

Since every multiple of $48$ is already included in $A \cup B$, we exclude them:

$450 - 18 = 432$

7. Final answer:

$\boxed{432}$

So the correct option is D.