How many ways can change be made for a quarter using standard U.S. coins?  (Don't count "1 quarter" as making change for a quarter.)
I can start by considering how many dimes I can use, and then for each possible number of dimes, how many nickels and pennies I can use. If I use 2 dimes, that leaves 5 cents to make with nickels and pennies. I can either use 1 nickel and 0 pennies, or 0 nickels and 5 pennies. So that's 2 ways to make change with 2 dimes. If I use 1 dime, that leaves 15 cents to make with nickels and pennies. I can use either 3 nickels and 0 pennies, or 2 nickels and 5 pennies, or 1 nickel and 10 pennies, or 0 nickels and 15 pennies. So that's 4 ways to make change with 1 dime. If I use 0 dimes, that leaves 25 cents to make with nickels and pennies. I can use either 5 nickels and 0 pennies, or 4 nickels and 5 pennies, or 3 nickels and 10 pennies, or 2 nickels and 15 pennies, or 1 nickel and 20 pennies, or 0 nickels and 25 pennies. So that's 6 ways to make change with 0 dimes. To get the total number of ways to make change for a quarter, I can add up the number of ways for each possible number of dimes: 2 + 4 + 6 = 12. 