Three are two urns Urn A has 3 distinct red balls and urin B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. Two number of ways in which this can be done is

Options

A. 36
B. 66
C. 108
D. 3

Answer

To solve the problem of transferring balls between two urns, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Balls in Each Urn**:
- Urn A contains 3 distinct red balls: R1, R2, R3.
- Urn B contains 9 distinct blue balls: B1, B2, B3, B4, B5, B6, B7, B8, B9.

2. **Choose 2 Balls from Urn A**:
- We need to select 2 balls from the 3 red balls in Urn A. The number of ways to do this is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.
- Thus, the number of ways to choose 2 balls from 3 is:
\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3 \]

3. **Choose 2 Balls from Urn B**:
- Next, we need to select 2 balls from the 9 blue balls in Urn B. Using the combination formula again:
- The number of ways to choose 2 balls from 9 is:
\[ \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

4. **Calculate the Total Number of Ways**:
- Since the selections from both urns are independent, we multiply the number of ways to choose from Urn A and Urn B:
\[ \text{Total ways} = \binom{3}{2} \times \binom{9}{2} = 3 \times 36 = 108 \]

5. **Conclusion**:
- The total number of ways in which 2 balls can be taken out from each urn and transferred to the other is **108**.

### Final Answer:
The number of ways in which this can be done is **108**.

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