1. Understand the process

We have:

• Urn A: $3$ distinct red balls
• Urn B: $9$ distinct blue balls

We must:

• choose $2$ balls from urn A,
• choose $2$ balls from urn B,
• then transfer them to the other urn.

So the number of ways depends only on the selections from each urn.

2. Choose 2 balls from urn A

From $3$ distinct balls, the number of ways to choose $2$ is

$\binom{3}{2} = 3$

3. Choose 2 balls from urn B

From $9$ distinct balls, the number of ways to choose $2$ is

$\binom{9}{2} = \frac{9\cdot 8}{2} = 36$

4. Total number of ways

These two choices are independent, so total ways:

$\binom{3}{2}\binom{9}{2} = 3 \times 36 = 108$

5. Check options

• A: $36$ ❌
• B: $66$ ❌
• C: $108$ ✅
• D: $3$ ❌

Therefore, the correct option is

$\boxed{108}$

So, Option C is correct.