How many different sums of money are possible using at least 3 coins from a collection consisting of one penny, one nickel, one dime, on quarter and one loonie?
This is a probability problem
5!/(3!2!) =10 different ways.
This question is not as easy as it first appears.
John found the number of ways to select exactly 3 coins from the 5 given. But it said at least 3 coins
so we could have 4 or all 5
number of ways to select
= C(5,3) + C(5,4) + C(5,5)
= 10 + 5 + 1 = 16
But we have not yet considered the value of these selections. Since it asked the number of different sums of money.
I can't see any other way than
listing all 16 combinations, and finding the value of each one.
The question is, "Are there any of the choices giving the same sum ?"
btw, since a "loonie" is a Canadian coins, I assume you re talking about Canadian coins. The teacher assigning this question should know that we have not used pennies for quite some time now.
To find out how many different sums of money are possible using at least 3 coins from the given collection, we can analyze the possible combinations.
Let's consider the coins one by one:
1. Penny (1 cent): This coin can be used in any combination as many times as we want.
2. Nickel (5 cents): This coin can be used in any combination as many times as we want.
3. Dime (10 cents): This coin can be used in any combination as many times as we want.
4. Quarter (25 cents): This coin can be used in any combination as many times as we want.
5. Loonie (1 dollar): This coin can also be used in any combination as many times as we want.
To find the number of different sums, we need to consider the combinations with at least 3 coins. We can start by listing all possible combinations of 3 coins:
- Penny, Nickel, Dime
- Penny, Nickel, Quarter
- Penny, Nickel, Loonie
- Penny, Dime, Quarter
- Penny, Dime, Loonie
- Penny, Quarter, Loonie
- Nickel, Dime, Quarter
- Nickel, Dime, Loonie
- Nickel, Quarter, Loonie
- Dime, Quarter, Loonie
This gives us a total of 10 different combinations.
Considering combinations with 4 coins, we can have:
- Penny, Nickel, Dime, Quarter
- Penny, Nickel, Dime, Loonie
- Penny, Nickel, Quarter, Loonie
- Penny, Dime, Quarter, Loonie
- Nickel, Dime, Quarter, Loonie
This gives us a total of 5 additional combinations.
Finally, considering combinations with all 5 coins (1 penny, 1 nickel, 1 dime, 1 quarter, 1 loonie), we have only 1 combination.
Therefore, the total number of different sums possible using at least 3 coins from the given collection is 10 + 5 + 1 = 16.
To find the number of different sums of money using at least 3 coins from the given collection, we can take a systematic approach.
First, let's look at the possible combinations of 3 coins. We have 5 coins in total, so the number of ways to select 3 coins is given by the binomial coefficient "5 choose 3," which can be calculated as follows:
5 choose 3 = 5! / (3!(5-3)!) = 5! / (3!2!) = (5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (2 * 1)) = 10.
Therefore, there are 10 different combinations of 3 coins we can select from the given collection.
Next, let's consider the combinations of 4 coins. Similarly, we can calculate the number of ways to select 4 coins using the binomial coefficient "5 choose 4:"
5 choose 4 = 5! / (4!(5-4)!) = 5! / (4!1!) = (5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (1)) = 5.
So, there are 5 different combinations of 4 coins.
Lastly, let's consider the combination of all 5 coins. There is only one way to select all the coins together.
Therefore, the total number of different sums of money using at least 3 coins is given by the sum of the combinations for 3, 4, and the combination of all 5 coins:
10 (combinations of 3 coins) + 5 (combinations of 4 coins) + 1 (combination of all 5 coins) = 16.
Hence, there are 16 different sums of money possible using at least 3 coins from the given collection.