## Thinking Mathematically (6th Edition)

To make sure we count all the ways, we'll list all possible combinations of all the different coins, sorted by the number of the most valuable coins spent. - (We can't use 3 dimes, as they're worth more than a quarter $(3*10>25)$. - We can use 2 dimes and 1 nickel $(2*10+5=25)$. - We can use 2 dimes and 5 pennies $(2*10+5*1=25)$ - We can use 1 dime and 3 nickels $(10+3*5=25)$ - We can use 1 dime, 2 nickels and 5 pennies $(10+2*5+5*1=25)$ - We can use 1 dime, 1 nickel, and 10 pennies $(10+5+10*1=25)$ - We can use 1 dime and 15 pennies $(10+15*1=25)$ - We can use 5 nickels $(5*5=25)$ - We can use 4 nickels and 5 pennies $(4*5+5*1=25)$ - We can use 3 nickels and 10 pennies $(3*5+10*1=25)$ - We can use 2 nickels and 15 pennies $(2*5+15*1=25)$ - We can use 1 nickel and 20 pennies $(5+20*1=25)$ - We can use 25 pennies $(25*1=25)$ This means we can make change for a quarter in a total of $12$ ways.