How many different sums of money can you make with three coins of different denominations?

Thanks

It would depend on the denominations.

If the denominations are coprime with each other, then the number of different sums would equal to the power set of the coins.
If |A|=3, |Power set (A)|=2^3.
That includes the amount $0, i.e. no coin used.

A notable example would be for denominations
$0.10, $0.20, $0.40
would give all amounts from $0 to 0.70, giving 2^3=8 different amounts.

However, if a denomination is the sum of the other two, then the number of different sums will be reduced.
Try for example $0.05, $0.10, $0.20.

To find the number of different sums of money that can be made with three coins of different denominations, we can approach this problem systematically. Let's assume the denominations of the three coins are A, B, and C (A < B < C).

Step 1: Find the maximum sum
The maximum sum of money that can be made is when all three coins are used. Therefore, the maximum sum would be A + B + C.

Step 2: Find the minimum sum
The minimum sum of money that can be made is when only one coin is used. Therefore, the minimum sum would be the denomination of the smallest coin, which is A.

Step 3: Find the number of different sums between the minimum and maximum sums
To find the number of different sums between the minimum and maximum sums, we need to consider the number of options we have for each coin. Assuming there are n denominations for each coin, we would have n options for the first coin, n-1 options for the second coin (since one denomination has already been selected for the first coin), and n-2 options for the third coin (since two denominations have already been selected for the first and second coins). Therefore, the total number of different sums would be (n) * (n-1) * (n-2).

Let's take an example:
Assume we have three coins with denominations of 1, 2, and 5.

Step 1: Maximum sum = 1 + 2 + 5 = 8
Step 2: Minimum sum = 1
Step 3: Number of different sums = (3) * (3-1) * (3-2) = 6

Hence, with three coins of different denominations, you can make six different sums of money.

To find the number of different sums of money that can be made with three coins of different denominations, we can use a simple method called brute force or list all possible combinations.

Let's assume the three coin denominations are A, B, and C.

To find all the possible sums, we can follow these steps:

1. Start with the smallest denomination coin, A.
- Count the number of possible sums using only coin A.
- In this case, there is only 1 possible sum: the value of coin A itself.

2. Move on to the second denomination coin, B.
- For each sum found using only coin A, add coin B to get new sums.
- Also count any new sums that are not already counted.
- In this step, if B is not equal to or less than the sum used in the previous step, then there will be more sums. Otherwise, it will be eliminated to avoid duplicates.

3. Finally, move on to the third denomination coin, C.
- For each sum found in the previous steps, add coin C to get new sums.
- Count any new sums that are not already counted.

The total count of sums obtained from the above steps will give you the number of different possible sums of money that can be made with the three coins of different denominations.

Let's illustrate this with an example:

Assume the three coin denominations are A = $1, B = $2, and C = $5.

1. Possible sums with coin A only: $1
2. Add coin B to the previous sums:

1 + 2 = $3
1 + 2 + 2 = $5

Total count: 2 (new sums - $3, $5)

3. Add coin C to the previous sums:

1 + 5 = $6
3 + 5 = $8

Total count: 2 (new sums - $6, $8)

Therefore, the number of different possible sums of money that can be made with three coins of different denominations is 2.

Please note that the number of different sums may vary depending on the specific denominations of the coins in question.