# Combinations of a penny, nickle, dime, and quarter

You have one penny, one nickle, one dime, and one quarter. How many different amounts of money can you make using one or more of these coins? Please help me! I'm having trouble! Im having trouble I know the obvious ones like 5,10,25,1,15,26,36,16.... Help? And I need to know the coin combinations too. I found 15 different ones! Anyone who can find more??

-
Have you tried to listed all the combinations? – Jack Mar 19 '12 at 3:34
@Esmeralda How many coins can you use in total? – azarel Mar 19 '12 at 3:34
@azarel..You can use just one of each.. – Esmeralda Mar 19 '12 at 3:39
And there are four coins – Esmeralda Mar 19 '12 at 3:45
Since there are exactly 16 subsets of the set of coins, one being empty, $15$ is an upper bound for the number of different amounts that can be obtained with $4$ coins; if you found $15$, you found them all. – Arturo Magidin Mar 19 '12 at 3:46

So, I think it is good to think one step at a time instead of trying to list off all the possibilities as they occur to you.

A good way to do this is to consider that since you have 4 total coins (1 penny, 1 nickel, 1 dime, and 1 quarter... I am understanding correctly that you have 4 total coins, right?) and you can choose 1 or more coins, you will be choosing 1,2,3 or 4 coins.

As you noted, there are some obvious ones, and I would say the most obvious are when you just choose one coin. You got all of those on your list: 1, 5, 10, 25.

Next, how many different ways can you choose 2 coins? Have you learned a formula for permutations and combinations? If not, just think systematically about how you would choose two coins from those 4. First take a penny and then how many choices do you have to get your second coin? Then you would choose the nickel and note how many choices you have for your second coin from there. Do you notice a pattern? Also take note that if you choose a penny and then a nickel, that is the same amount as choosing a nickel then a penny, so some of your choices will, in reality, be the same amount.

Can you proceed from here? Figure out the ways you can choose 3, and then how many ways can you choose 4?

And I should add, there are total ways to choose the coins, though 1 of those is to choose no coins. So the 15 amounts you have are all of them, but it's good to think about the problem systematically if you want to quickly write down all the combinations and start to see the pattern/formula to solve this type of problem in general.

-
@rar- thank you! I see the pattern! I'll do my best and let you know if I get it right or not (: – Esmeralda Mar 19 '12 at 3:47
I checked my work and still got 15 and acorrding to Arturo and your help I'm confident that I got it right!^__^ – Esmeralda Mar 19 '12 at 3:51
Nice answer. I wish all of the high school math textbooks were written this way. – scaaahu Mar 19 '12 at 4:05
@scaaahu Thanks. I never know if I'm writing things that make sense when I answer questions. – user23784 Mar 19 '12 at 4:15