How many ways can you make change for a dollar using only quarters, dimes, and nickels, if you use at most three nickels?
all of these are wrong, along with many others:
- 3 quarters, 2 dimes
- 3 quarters, 1 dime, 2 nickels
- 3 quarters, 3 nickels
- 2 quarters, 4 dimes
the actual ways to do it are:
only quarters:
4 quarters
only quarters and dimes:
2 quarters, 5 dimes
10 dimes
quarters, dimes, nickels (at most 3):
3 quarters, 2 dimes, 1 nickel
3 quarters, 1 dime, 3 nickels
2 quarters, 4 dimes, 2 nickels
1 quarter, 7 dimes, 1 nickel
1 quarter, 6 dimes, 3 nickels
So there are only 8 ways to do it
I apologize for the incorrect response. You are correct.
The correct ways to make change for a dollar using only quarters, dimes, and nickels, with at most three nickels, are as follows:
- Only quarters: 4 quarters
- Only quarters and dimes: 2 quarters, 5 dimes and 10 dimes
- Quarters, dimes, nickels (at most 3):
3 quarters, 2 dimes, 1 nickel
3 quarters, 1 dime, 3 nickels
2 quarters, 4 dimes, 2 nickels
1 quarter, 7 dimes, 1 nickel
1 quarter, 6 dimes, 3 nickels
So, there are indeed 8 ways to make change for a dollar using only quarters, dimes, and nickels, with at most three nickels.
To approach this problem, we can break it down step by step.
Step 1: Calculate the number of ways to make change using only quarters and dimes.
We can make change using only quarters and dimes in the following ways:
- 4 quarters
- 3 quarters, 2 dimes
- 3 quarters, 1 dime, 2 nickels
- 3 quarters, 3 nickels
- 2 quarters, 4 dimes
- 2 quarters, 3 dimes, 2 nickels
- 2 quarters, 2 dimes, 4 nickels
- 2 quarters, 1 dime, 6 nickels
- 2 quarters, 8 nickels
- 2 quarters, 2 nickels
- 1 quarter, 6 dimes
- 1 quarter, 5 dimes, 2 nickels
- 1 quarter, 4 dimes, 4 nickels
- 1 quarter, 3 dimes, 6 nickels
- 1 quarter, 2 dimes, 8 nickels
- 1 quarter, 1 dime, 10 nickels
- 1 quarter, 12 nickels
- 1 quarter, 2 nickels
- 0 quarters, 10 dimes
- 0 quarters, 9 dimes, 2 nickels
- 0 quarters, 8 dimes, 4 nickels
- 0 quarters, 7 dimes, 6 nickels
- 0 quarters, 6 dimes, 8 nickels
- 0 quarters, 5 dimes, 10 nickels
- 0 quarters, 4 dimes, 12 nickels
- 0 quarters, 3 dimes, 14 nickels
- 0 quarters, 2 dimes, 16 nickels
- 0 quarters, 1 dime, 18 nickels
- 0 quarters, 20 nickels
So there are 29 ways to make change using only quarters and dimes.
Step 2: Calculate the number of ways to make change using quarters, dimes, and nickels.
To do this, we need to consider the combinations of nickels that we can add to the 29 ways we calculated in Step 1 while ensuring that we use at most three nickels.
If we have 0 nickels, we already accounted for these combinations in Step 1, so we do not need to add anything here.
If we have 1 nickel, we can add it to each of the 29 combinations calculated in Step 1.
If we have 2 nickels, we can add them to each of the 29 combinations calculated in Step 1. Since we have two nickels, we can add two nickels to any of the combinations from Step 1. This will give us additional combinations:
- 2 dimes, 6 nickels
- 1 dime, 8 nickels
- 0 quarters, 4 dimes, 6 nickels
- 0 quarters, 3 dimes, 8 nickels
- 0 quarters, 2 dimes, 10 nickels
- 0 quarters, 1 dime, 12 nickels
- 0 quarters, 14 nickels
If we have 3 nickels, we can add them to each of the 29 combinations calculated in Step 1. Since we have three nickels, we can add three nickels to any of the combinations from Step 1. This will give us additional combinations:
- 1 dime, 9 nickels
- 0 quarters, 4 dimes, 7 nickels
- 0 quarters, 3 dimes, 9 nickels
- 0 quarters, 2 dimes, 11 nickels
- 0 quarters, 1 dime, 13 nickels
- 0 quarters, 15 nickels
So there are 29 combinations from Step 1, 29 combinations with 1 nickel, 7 combinations with 2 nickels, and 6 combinations with 3 nickels.
Total number of ways to make change for a dollar using only quarters, dimes, and nickels, with at most three nickels, is 29 + 29 + 7 + 6 = 71.