Change for a Dollar Date: 01/22/2001 at 17:35:06 From: maria E. Lacey Subject: Coins used to change one dollar Could you please tell me the two hundred plus ways in which coins are used to make change for one dollar? Date: 01/23/2001 at 11:43:55 From: Doctor Greenie Subject: Re: Coins used to change one dollar Hi, Maria - It wouldn't do you any good for us to tell you the 292 different ways to make change for a dollar. We are here to teach mathematics - that is, to help you learn ways to answer questions like this for yourself. I will provide a sketch below of how I would go about enumerating all of the different ways to make change for a dollar. You can also find several pages in the Dr. Math archives where this problem is discussed (using slightly different approaches). To find these pages, click on the "Search the Archives" link on the main Dr. Math page and enter "change for a dollar" as the phrase to look for. (Be sure to click on the button that makes the search engine look for the entire phrase instead of the individual words.) The key to answering this question correctly and completely is to organize your list of ways to make change for a dollar in such a way that you make sure that you (a) count every possibility, and (b) don't count any possibilities more than once. With this goal in mind, there are two key features to my approach to this problem: (1) Break the problem into smaller parts. I will enumerate all the possible combinations I can have for the numbers of half dollars and quarters; then for each of those combinations I will enumerate the number of ways I can make up the remaining part of a dollar using dimes, nickels, and pennies. (2) Keep the list in either strictly ascending or strictly descending order in terms of the numbers of coins of different denominations. I personally prefer to order my list with the largest numbers of largest-denomination coins at the top. I hope this will be clear from my outline below. So here is a list of all the possible combinations I can have of half dollars and quarters. The list also shows, for each combination of half dollars and quarters, the remaining amount to made up using dimes, nickels, and pennies. remaining half amount dollars quarters (cents) ------------------------------- 2 0 0 1 2 0 1 1 25 1 0 50 0 4 0 0 3 25 0 2 50 0 1 75 0 0 100 Notice the arrangement of this list. The first entry is the one that uses the maximum number (2) of the largest-denomination coin (half dollars). The next three entries are those that use the next largest number (1) of half dollars; these three entries are arranged with the one using the largest number of quarters (2) first and the smallest number of quarters (0) last. Similarly, the last five entries use the smallest number (0) of half dollars; and they are arranged with the entry having the largest number of quarters (4) first and the smallest number of quarters (0) last. To complete the answer to the problem, you now need to find the numbers of different ways to make either 0, 25, 50, 75, or 100 cents using only dimes, nickels, and pennies. Once you have done that, you can find the total number of ways to make change for a dollar by adding a column to the table above showing the number of ways to make change for a dollar using each of the possible combinations of half dollars and quarters. Then your table will look like this: remaining number of ways to make half amount remaining amount using dollars quarters (cents) dimes, nickels, and pennies ------------------------------------------------------------- 2 0 0 1 1 2 0 1 1 1 25 12 1 0 50 36 0 4 0 1 0 3 25 12 0 2 50 36 0 1 75 72 0 0 100 121 ----- Total number of ways to make 292 change for a dollar I have filled in the numbers in the last column for you. Clearly if the amount to be made up using dimes, nickels, and pennies is 0 (i.e., the whole dollar is made up using half dollars and/or quarters) then there is exactly one way to make up the remaining "0 cents": 0 dimes, 0 nickels, and 0 pennies. To determine the other numbers in the last column, you can make an organized list of all the possibilities, as I have done for the first part of the problem above. For example, below is an outline of the organized list showing that there are 36 ways to make change for 50 cents using dimes, nickels, and pennies. In this list, I have not listed all the possible combinations. Instead, I have only indicated, for each possible number of dimes, the possible numbers of nickels I can use. I can do this because, since any amount of money can be made using all pennies, I am able to choose any number of nickels and make up the remaining amount with pennies. Notice again the arrangement of the entries, with the largest possible number of dimes first and the smallest possible number of dimes last. Number of Ways to Make Change for 50 Cents Using Dimes, Nickels, Pennies: choices for dimes number of nickels number of choices ------------------------------------------------- 5 0 1 4 0, 1, or 2 3 3 0 to 4 5 2 0 to 6 7 1 0 to 8 9 0 0 to 10 11 ---- Total number of ways to make 36 change for 50 cents using dimes, nickels, and pennies See if you can build the tables showing that the numbers of ways to make change for 25 cents, 75 cents, and 100 cents using dimes, nickels, and pennies are as shown in the table above (12, 72, and 121, respectively). When you have done so, you will have enumerated all 292 possible ways for making change for a dollar. Your original question asked for us to show you all the possible ways to make change for a dollar. As you can see, doing so would be very tedious. But at this point, you should know what the 292 different ways are, without having to write them out. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/