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. 17
• B. 16
• C. 18
• D. 15
• E. 19

Hint:

When counting currency combinations, break into cases based on largest denomination to simplify.

Answer 1

The Correct Option is A

To determine the number of ways to pay a bill of 107 Misos using denominations of 1 Miso, 10 Misos, and 50 Misos, we use a combinatorial approach. Let's denote:

• x: number of 1 Miso coins 
• y: number of 10 Miso coins
• z: number of 50 Miso coins

The equation to satisfy is:

x + 10y + 50z = 107

We need to find all possible non-negative integer solutions to this equation. We will consider the possible values of z (number of 50 Miso coins) first:

1. If z = 2:
   • The amount from 50 Misos is 100 Misos.
   • We need 107 - 100 = 7 Misos.
   • Equation becomes: x + 10y = 7.
   • Possible solution: (x, y) = (7, 0).
2. If z = 1:
   • The amount from 50 Misos is 50 Misos.
   • We need 107 - 50 = 57 Misos.
   • Equation becomes: x + 10y = 57.
   • Possible solutions for y are 0 to 5:
     • For y = 0: x = 57.
     • For y = 1: x = 47.
     • For y = 2: x = 37.
     • For y = 3: x = 27.
     • For y = 4: x = 17.
     • For y = 5: x = 7.
3. If z = 0:
   • We need the full amount: 107 Misos.
   • Equation becomes: x + 10y = 107.
   • Possible solutions for y are 0 to 10:
     • For y = 0: x = 107.
     • For y = 1: x = 97.
     • For y = 2: x = 87.
     • For y = 3: x = 77.
     • For y = 4: x = 67.
     • For y = 5: x = 57.
     • For y = 6: x = 47.
     • For y = 7: x = 37.
     • For y = 8: x = 27.
     • For y = 9: x = 17.
     • For y = 10: x = 7.

Hence, the total number of solutions is 1 (from 50 Miso = 100) + 6 (from 50 Miso = 50) + 11 (from no 50 Misos), which gives us a total of 17 ways to pay the bill of 107 Misos.