Key Concepts and Formulas

• Combinations ($^nC_k$ or $\binom{n}{k}$): The number of ways to choose $k$ objects from a set of $n$ distinct objects, where order doesn't matter, is given by: $^nC_k = \binom{n}{k} = \frac{n!}{k!(n-k)!}$
• Multiplication Principle: If there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both.

Step-by-Step Solution

1. Understand the Problem and Identify Independent Events

We have two urns, A and B. We are selecting 2 balls from each urn. The selection from urn A does not affect the selection from urn B, so these are independent events. We will use combinations to count the number of ways to select balls from each urn, and then the multiplication principle to find the total number of ways.

2. Calculate the Number of Ways to Select Balls from Urn A

• Given: Urn A contains 3 distinct red balls.
• Task: Choose 2 balls from Urn A.
• Since the balls are distinct and order doesn't matter, we use combinations.
• Applying the formula: Here, $n=3$ and $k=2$. $^3C_2 = \frac{3!}{2!(3-2)!} = \frac{3!}{2!1!} = \frac{3 \times 2 \times 1}{(2 \times 1)(1)} = \frac{6}{2} = 3$ There are 3 ways to choose 2 balls from Urn A.

3. Calculate the Number of Ways to Select Balls from Urn B

• Given: Urn B contains 9 distinct blue balls.
• Task: Choose 2 balls from Urn B.
• Since the balls are distinct and order doesn't matter, we use combinations.
• Applying the formula: Here, $n=9$ and $k=2$. $^9C_2 = \frac{9!}{2!(9-2)!} = \frac{9!}{2!7!} = \frac{9 \times 8 \times 7!}{(2 \times 1) \times 7!} = \frac{9 \times 8}{2} = \frac{72}{2} = 36$ There are 36 ways to choose 2 balls from Urn B.

4. Apply the Multiplication Principle

Since the selections from Urn A and Urn B are independent events, we multiply the number of ways for each event to find the total number of ways: Total ways = (Ways to choose from Urn A) $\times$ (Ways to choose from Urn B) Total ways = $3 \times 36 = 108$

Common Mistakes & Tips

• Combinations vs. Permutations: Remember to use combinations when the order of selection doesn't matter. Using permutations would give the wrong answer.
• "Distinct" is Important: Pay attention to the word "distinct." If the balls were identical, we would need a different approach.
• Focus on the Selection: The problem asks about the number of ways to select the balls. The transfer part is irrelevant to the calculation.

Summary

We used combinations to calculate the number of ways to select 2 balls from each urn, and then the multiplication principle to combine these results. The total number of ways is $3 \times 36 = 108$.

The final answer is $\boxed{108}$, which corresponds to option (C).