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?
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?
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When counting currency combinations, break into cases based on largest denomination to simplify.
Updated On: Jul 31, 2025
- 17
- 16
- 18
- 15
- 19
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The Correct Option is A
Solution and Explanation
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:
- 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).
- 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.
- 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.
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