I Have 5 coins: a penny, a dime, a nickel, a quarter, a loonie. How many different amounts of money could I pay using any combinations of these coins?
I can use the penny in 2 ways, either take it or not take it.
I can use the dime in 2 ways, either take it or not take it.
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So the number of ways is
2x2x2x2x2 = 32
but that would include the case of not taking any of the coins, and we want to have at least some kind of value
so the number of ways is 32-1 = 31
Canadian, eh?
To find out how many different amounts of money you can make using different combinations of these coins, you need to consider all possible subsets of the coins. Here's how you can calculate it step by step:
1. Start by excluding all the coins. This will give you one possibility, which is not having any coins.
- Zero coins: $0.00
2. Now, consider each coin individually. You can choose to include or exclude each coin.
- One coin:
- Penny: $0.01
- Dime: $0.10
- Nickel: $0.05
- Quarter: $0.25
- Loonie: $1.00
3. Next, consider combinations of two coins. There are 10 possible combinations:
- Penny + Dime: $0.01 + $0.10 = $0.11
- Penny + Nickel: $0.01 + $0.05 = $0.06
- Penny + Quarter: $0.01 + $0.25 = $0.26
- Penny + Loonie: $0.01 + $1.00 = $1.01
- Dime + Nickel: $0.10 + $0.05 = $0.15
- Dime + Quarter: $0.10 + $0.25 = $0.35
- Dime + Loonie: $0.10 + $1.00 = $1.10
- Nickel + Quarter: $0.05 + $0.25 = $0.30
- Nickel + Loonie: $0.05 + $1.00 = $1.05
- Quarter + Loonie: $0.25 + $1.00 = $1.25
4. Continuing this process, consider combinations of three coins. There are 10 possible combinations:
- Penny + Dime + Nickel: $0.01 + $0.10 + $0.05 = $0.16
- Penny + Dime + Quarter: $0.01 + $0.10 + $0.25 = $0.36
- Penny + Dime + Loonie: $0.01 + $0.10 + $1.00 = $1.11
- Penny + Nickel + Quarter: $0.01 + $0.05 + $0.25 = $0.31
- Penny + Nickel + Loonie: $0.01 + $0.05 + $1.00 = $1.06
- Penny + Quarter + Loonie: $0.01 + $0.25 + $1.00 = $1.26
- Dime + Nickel + Quarter: $0.10 + $0.05 + $0.25 = $0.40
- Dime + Nickel + Loonie: $0.10 + $0.05 + $1.00 = $1.15
- Dime + Quarter + Loonie: $0.10 + $0.25 + $1.00 = $1.35
- Nickel + Quarter + Loonie: $0.05 + $0.25 + $1.00 = $1.30
5. Lastly, consider combinations of four coins. There are 5 possible combinations:
- Penny + Dime + Nickel + Quarter: $0.01 + $0.10 + $0.05 + $0.25 = $0.41
- Penny + Dime + Nickel + Loonie: $0.01 + $0.10 + $0.05 + $1.00 = $1.16
- Penny + Dime + Quarter + Loonie: $0.01 + $0.10 + $0.25 + $1.00 = $1.36
- Penny + Nickel + Quarter + Loonie: $0.01 + $0.05 + $0.25 + $1.00 = $1.31
- Dime + Nickel + Quarter + Loonie: $0.10 + $0.05 + $0.25 + $1.00 = $1.40
6. Finally, consider all five coins together, which is one combination:
- Penny + Dime + Nickel + Quarter + Loonie: $0.01 + $0.10 + $0.05 + $0.25 + $1.00 = $1.41
Therefore, combining all the possibilities from each step, you can make a total of 1 + 5 + 10 + 10 + 5 + 1 = 32 different amounts of money using any combination of these coins.