Suppose you have an unlimited number pennies, nickels, dimes, and quarters. Determine the number of ways to make 30 cents using these coins.
We use cases to organize our work, based first on the number of quarters and then the number of dimes. First note that the number of quarters must be 0 or 1 , since 2 quarters would be too much. This gives 2 cases:

- 1 quarter: There are 2 possibilities: a quarter and a nickel or a quarter and 5 pennies.
- $\mathbf{0}$ quarters: If we don't use quarters, we can use at most 2 dimes, so we can make 3 subcases based on the number of dimes:
- $\mathbf{2}$ dimes: We need to make 10 cents more using nickels or pennies. We could use 0,1 , or 2 nickels, so there are 3 possibilities.
- 1 dime: We need to make 20 cents more using nickels or pennies. We could use $0,1,2,3$, or 4 nickels, so there are 5 possibilities.
- $\mathbf{0}$ dimes: We need to make 30 cents more using nickels and pennies. We could use $0,1,2,3,4,5$, or 6 nickels, so there are 7 possibilities.

Putting this together, we get a total of $2+3+5+7=17$ possibilities.