A purse contains three 10 paise, three 50 paise and ten 1 rupee coins. If three coins are selected at random, then the probability that the total amount is 2 rupee is

Options

A. `(2)/(5)`
B. `(3)/(56)`
C. `(1)/(4)`
D. `(1)/(8)`

Answer

To solve the problem, we need to find the probability that the total amount of three randomly selected coins from a purse containing three 10 paise coins, three 50 paise coins, and ten 1 rupee coins equals 2 rupees.

### Step 1: Understand the Coin Denominations
- We have:
- 3 coins of 10 paise
- 3 coins of 50 paise
- 10 coins of 1 rupee

### Step 2: Total Number of Coins
- Total coins = 3 (10 paise) + 3 (50 paise) + 10 (1 rupee) = 16 coins.

### Step 3: Calculate Total Ways to Choose 3 Coins
- The total number of ways to choose 3 coins from 16 coins is given by the combination formula:

\[ \text{Total ways} = \binom{16}{3} = \frac{16 \times 15 \times 14}{3 \times 2 \times 1} = 560 \]

### Step 4: Identify Combinations that Sum to 2 Rupees
To achieve a total of 2 rupees (200 paise), we can consider the following combinations of coins:
1. **2 coins of 50 paise and 1 coin of 1 rupee**:
- Value from 2 coins of 50 paise = 100 paise
- Value from 1 coin of 1 rupee = 100 paise
- Total = 100 + 100 = 200 paise (2 rupees)

### Step 5: Calculate the Number of Favorable Outcomes
- The number of ways to select 2 coins from 3 (50 paise) and 1 coin from 10 (1 rupee) is:

\[ \text{Ways to choose 2 coins of 50 paise} = \binom{3}{2} = 3 \]
\[ \text{Ways to choose 1 coin of 1 rupee} = \binom{10}{1} = 10 \]

- Therefore, the total number of favorable outcomes is:

\[ \text{Favorable outcomes} = 3 \times 10 = 30 \]

### Step 6: Calculate the Probability
- The probability \( P \) that the total amount is 2 rupees is given by the ratio of favorable outcomes to total outcomes:

\[ P = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{30}{560} = \frac{3}{56} \]

### Final Answer
Thus, the probability that the total amount is 2 rupees when selecting 3 coins at random is:

\[ \boxed{\frac{3}{56}} \]