Combinations of a penny, nickel, dime, and quarter You have one penny, one nickel, 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! I'm having trouble I know the obvious ones like $5,10,25,1,15,26,36,16,\ldots$ Help? And I need to know the coin combinations too. I found $15$ different ones! Anyone who can find more??
 A: 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.
A: As no combinations of your coins add up the the same amount the number of amounts is the number of subsets of coins.  With $4$ items there are $2^4=16$ subsets, so you can make $16$ different totals.  If you don't want to count $0$, subtract $1$.
