PROBLEM: Coins
In how many ways can 19 coins equal exacly one dollar?
SOLUTION:
Before we even attempted this problem, we had to look at how many coins within each area, would equal one dollar. The outcome was:
The easiest to see here was that there was not 19 coins in any one particular section/pieces of money that would or could equal one dollar. The next idea that came to mind was mixing and matching 19 coins to get to one dollar.
When I began working these combinations out on paper, the first denomination that I used was 5 cent pieces. I figured it seemed like the easiset to use since I was so close to making one dollar out of 19 coins.
Look at my list:





























































Just from my studies of the above numbers, it looks as if the number of coins used in the one cent column is only a 5, 10, or 15, which means it could only be using multiples of 5 (but only these three numbers were used).
For some reason, I could not figure out what to do next, especially after seeing that multiples of 5 jumped out at me for the one cent column. I began looking to see if there was some sort of pattern for the other coins. The only thing we could see was that there could not be more than 18 nickels. There also could not be more than one 50 cent piece (because two would automatically equal $1), 3 quarters (same as the reason before), 9 dimes, and 15 pennies.
Now I still did not know what to do next. Shae decided that since we had gotten that far, why not just multiply the total number of the "not more thans" from each column to get a feel for what the total may indeed be. I told here that this number seems big especially since, we are only finding 19 coins whose total value added together was $1. That answer was: 1*3*9*18*3 = 1458. What are we going to do with this number I kept asking her. She only said it was a reference for us.
I had another look at the table above after my discussion with Shae. I noticed that there were two ways to get $1 using only two different types of coins, three ways to get $1 using three different types of coins, and four ways to get $1 using 4 different types of coins. Is this a coincidence?
Are there any other ways other than the ones listed above? I would certainly like to know. Email me with any info.