Question

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?

• A. 18
• B. 16
• C. 17
• D. 15
• E. 19

Hint:

Fix the number of 50-Miso notes first, then count the choices for the 10-Miso notes; the 1-Miso coins fill up the rest.

Answer 1

The Correct Option is A

Step 1: Set up:
Let \( a, b, c \) be the numbers of 50, 10 and 1 Miso pieces. We need \( 50a + 10b + c = 107 \) with \( a, b, c \ge 0 \) integers. Once \( a \) and \( b \) are chosen with \( 50a+10b \le 107 \), the number \( c \) is fixed.

Step 2: Count by the number of 50-Miso notes:

• \( a = 0 \): \( 10b \le 107 \), so \( b = 0,1,\ldots,10 \): 11 ways.
• \( a = 1 \): \( 10b \le 57 \), so \( b = 0,1,\ldots,5 \): 6 ways.
• \( a = 2 \): \( 10b \le 7 \), so \( b = 0 \): 1 way.

Step 3: Add:
\( 11 + 6 + 1 = 18 \).

Final Answer:
The correct option is (1): 18.