Vinod have 20 rupees. He bought 1, 2, 5 rupee stamps. They are different in numbers by the reason of no change, the shop keeper gives 3 one rupee stamps. So how many stamps Vinod have ?

Options

A. A) 10
B. B) 18
C. C) 12
D. D) 15

Answer

To solve the problem step by step, we will analyze the given information and derive the number of stamps Vinod has.

### Step 1: Understand the total amount Vinod has
Vinod has a total of 20 rupees.

### Step 2: Identify the stamps he received
The shopkeeper gives Vinod 3 one-rupee stamps. Therefore, he has already spent 3 rupees.

### Step 3: Calculate the remaining amount
After receiving the 3 one-rupee stamps, the amount Vinod has left is:
\[ 20 - 3 = 17 \text{ rupees} \]

### Step 4: Define the variables for the number of stamps
Let:
- \( K \) = number of 5-rupee stamps
- \( L \) = number of 2-rupee stamps

### Step 5: Set up the equation based on the remaining amount
The total value of the 2-rupee and 5-rupee stamps must equal the remaining amount:
\[ 2L + 5K = 17 \]

### Step 6: Consider the condition of different numbers of stamps
We know that the number of 1-rupee, 2-rupee, and 5-rupee stamps must be different. Since Vinod has 3 one-rupee stamps, \( K \) (number of 5-rupee stamps) cannot be 3.

### Step 7: Find possible values for \( K \) and \( L \)
We can try different values for \( K \) to find corresponding values for \( L \):

1. If \( K = 1 \):
\[ 2L + 5(1) = 17 \implies 2L + 5 = 17 \implies 2L = 12 \implies L = 6 \]
(This gives us 1 five-rupee stamp and 6 two-rupee stamps)

2. If \( K = 2 \):
\[ 2L + 5(2) = 17 \implies 2L + 10 = 17 \implies 2L = 7 \implies L = 3.5 \]
(Not possible since \( L \) must be an integer)

3. If \( K = 3 \):
\[ 2L + 5(3) = 17 \implies 2L + 15 = 17 \implies 2L = 2 \implies L = 1 \]
(This gives us 3 five-rupee stamps, but violates the condition of different numbers)

4. If \( K = 0 \):
\[ 2L + 5(0) = 17 \implies 2L = 17 \implies L = 8.5 \]
(Not possible since \( L \) must be an integer)

### Step 8: Confirm the valid combination
From our calculations, the only valid combination is:
- 3 one-rupee stamps
- 1 five-rupee stamp
- 6 two-rupee stamps

### Step 9: Calculate the total number of stamps
Total number of stamps Vinod has:
\[ 3 \text{ (one-rupee)} + 1 \text{ (five-rupee)} + 6 \text{ (two-rupee)} = 10 \text{ stamps} \]

### Final Answer
Vinod has a total of **10 stamps**.

---