Suppose you have a currency, named Miso, in three denominations: 1 Miso, 10 Misos and 50 Misos. In how many ways can you pay a bill of 107 Misos?

Options

A. 19
B. 17
C. 16
D. 18

Answer

To solve the problem of how many ways we can pay a bill of 107 Misos using denominations of 1 Miso, 10 Misos, and 50 Misos, we can break down the problem as follows:

### Step-by-Step Solution:

1. **Define the Variables**:
Let:
- \( x \) = number of 1 Misos used
- \( y \) = number of 10 Misos used
- \( z \) = number of 50 Misos used

We need to satisfy the equation:
\[ x + 10y + 50z = 107 \]

2. **Determine the Possible Values for \( z \)**:
Since the maximum value of \( z \) (the number of 50 Misos) can be determined by how many times 50 can fit into 107:
- If \( z = 0 \): The equation becomes \( x + 10y = 107 \)
- If \( z = 1 \): The equation becomes \( x + 10y = 57 \)
- If \( z = 2 \): The equation becomes \( x + 10y = 7 \)
- \( z \) cannot be 3 or more since \( 50 \times 3 = 150 \) exceeds 107.

3. **Case 1: When \( z = 0 \)**:
The equation simplifies to:
\[ x + 10y = 107 \]
Here, \( y \) can take values from 0 to 10 (as \( 10 \times 10 = 100 \) and \( 107 - 100 = 7 \) allows for 7 more 1 Misos). Therefore, the possible values of \( y \) are:
- \( y = 0 \) to \( y = 10 \) (11 possibilities)

4. **Case 2: When \( z = 1 \)**:
The equation simplifies to:
\[ x + 10y = 57 \]
Here, \( y \) can take values from 0 to 5 (as \( 10 \times 5 = 50 \) and \( 57 - 50 = 7 \) allows for 7 more 1 Misos). Therefore, the possible values of \( y \) are:
- \( y = 0 \) to \( y = 5 \) (6 possibilities)

5. **Case 3: When \( z = 2 \)**:
The equation simplifies to:
\[ x + 10y = 7 \]
Here, \( y \) can only be 0 (as \( 10 \times 1 = 10 \) exceeds 7). Therefore, the possible values of \( y \) are:
- \( y = 0 \) (1 possibility)

6. **Calculate the Total Number of Ways**:
Now, we add the number of possibilities from each case:
- From Case 1: 11 ways
- From Case 2: 6 ways
- From Case 3: 1 way

Total ways = \( 11 + 6 + 1 = 18 \)

### Final Answer:
Thus, the total number of ways to pay a bill of 107 Misos is **18 ways**.

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