In a purse there are 9 five paisa coins and one rupee coin. In another purse there are all 10 five paisa coins. 9 coins are taken from the former and put into the second and then 9 coins are taken from the latter and put into the first. What is the chance that rupee coin is still in the first purse

Options

A. `1//9`
B. `10//19`
C. `5//19`
D. none

Answer

To solve the problem, we need to determine the probability that the rupee coin remains in the first purse after the transfers of coins.

### Step-by-Step Solution:

1. **Initial Setup**:
- In the first purse (Purse A), there are 9 five paisa coins and 1 rupee coin (total = 10 coins).
- In the second purse (Purse B), there are 10 five paisa coins (total = 10 coins).

2. **Transfer Coins from Purse A to Purse B**:
- We take 9 coins from Purse A and put them into Purse B.
- The possible scenarios for the coins transferred from Purse A are:
- Case 1: The rupee coin is included in the 9 coins transferred.
- Case 2: The rupee coin is not included in the 9 coins transferred.

3. **Case Analysis**:
- **Case 1**: The rupee coin is transferred to Purse B.
- If the rupee coin is transferred, it cannot return to Purse A. Therefore, the probability of the rupee coin being in Purse A after the transfers is 0.

- **Case 2**: The rupee coin is not transferred to Purse B.
- If the rupee coin is not transferred, then all 9 coins transferred from Purse A to Purse B are five paisa coins.
- The composition of Purse B after the transfer will be:
- 10 five paisa coins (originally in Purse B) + 9 five paisa coins (transferred from Purse A) = 19 five paisa coins.
- Now, we need to transfer 9 coins back from Purse B to Purse A.

4. **Calculating the Probability**:
- The total number of ways to choose 9 coins from 19 coins in Purse B is given by \( \binom{19}{9} \).
- The number of ways to choose 9 coins such that the rupee coin is included (which means we choose 8 more coins from the remaining 18 five paisa coins) is given by \( \binom{18}{8} \).

5. **Probability that the Rupee Coin is in Purse A**:
- The probability that the rupee coin is included in the 9 coins taken from Purse B is:
\[ P(\text{Rupee coin in A}) = \frac{\binom{18}{8}}{\binom{19}{9}} \]
- Using the property of combinations:
\[ \binom{19}{9} = \frac{19!}{9! \cdot 10!} \quad \text{and} \quad \binom{18}{8} = \frac{18!}{8! \cdot 10!} \]
- Thus,
\[ P(\text{Rupee coin in A}) = \frac{\frac{18!}{8! \cdot 10!}}{\frac{19!}{9! \cdot 10!}} = \frac{18! \cdot 9!}{19! \cdot 8!} = \frac{9}{19} \]

6. **Final Probability**:
- The total probability that the rupee coin remains in Purse A after both transfers is:
\[ P(\text{Rupee coin in A}) = 0 \cdot P(\text{Case 1}) + \frac{9}{19} \cdot P(\text{Case 2}) = \frac{9}{19} \]

### Conclusion:
The probability that the rupee coin is still in the first purse is \( \frac{10}{19} \).