how many different sums of money can you make with a penny, a dime, a one-dollar coin, and a two-dollar coin?
well the answer in the textbook is 15? :S i don't know, it's in the unit of combinations and the binomial theorem
2^n-1=2^4-1=15
2 is always the base (you either pick it or you don't) and 4 because there are 4 options and subtract one because you exclude the empty or null set.
To find out how many different sums of money you can make with a penny ($0.01), a dime ($0.10), a one-dollar coin ($1.00), and a two-dollar coin ($2.00), we can use a method called exhaustive counting. We will consider all possible combinations of these coins.
One way to approach this is by listing out all the possible coin combinations and adding up their values. Here's how you can do it:
1. Start with the highest denomination coin, the two-dollar coin ($2.00). You can either use it or not use it.
a. If you use the two-dollar coin, you have $2.00 as the total sum.
b. If you don't use the two-dollar coin, move on to the next coin.
2. Now consider the one-dollar coin ($1.00). Again, you can either use it or not use it in combination with the two-dollar coin (if you used it). If you didn't use the two-dollar coin, the one-dollar coin becomes the highest denomination coin.
a. If you use the one-dollar coin (in combination with or without the two-dollar coin), you have $1.00 or $3.00 as the total sum.
b. If you don't use the one-dollar coin, move on to the next coin.
3. Next, consider the dime ($0.10). Again, you have two options: use it or don't use it in combination with the coins you've used so far.
a. If you use the dime, you can combine it with the two-dollar and/or one-dollar coins (if they were used). This gives you the following combination possibilities:
- $0.10 (only the dime)
- $2.10 (two dollars and a dime)
- $1.10 (one dollar and a dime)
- $3.10 (three dollars and a dime)
b. If you don't use the dime, move on to the next coin.
4. Finally, we have the penny ($0.01). Again, you can either use it or not use it in combination with the coins you've used so far.
a. If you use the penny, you can combine it with any of the coin combinations we've determined so far. This gives you additional possibilities:
- $0.01 (only the penny)
- $2.01 (two dollars and a penny)
- $1.01 (one dollar and a penny)
- $3.01 (three dollars and a penny)
- $0.11 (a dime and a penny)
- $2.11 (two dollars, a dime, and a penny)
- $1.11 (one dollar, a dime, and a penny)
- $3.11 (three dollars, a dime, and a penny)
b. If you don't use the penny, you have already exhausted all possible combinations.
By considering all combinations, you can observe that there are a total of 12 different sums of money that can be made using a penny, a dime, a one-dollar coin, and a two-dollar coin.