Thinking Mathematically (6th Edition)

Let's look at the coins from quarters to nickels. $2*25 = 50$, which is more than $45$, therefore, we can only spend 1 or 0 quarters. Let's first count all the cases in which we spend one quarter. We need to spend the remaining $45-25=20$ cents. We can do so by: - Spending 2 dimes, at 10 cents each, to make 20. - Spending 1 dime, and 2 nickels, to make $10+5+5=20$. - Spending 4 nickels to make $5*4=20$. This is 3 ways in total to spend $45$ if we spend one quarter. In the cases where we spend no quarters: - (We don't have 4 dimes, so we have to spend 3 at most) - We can spend 3 dimes and 3 nickels, for a total of $3*10+3*5=45$ - We can spend 2 dimes and 5 nickels, for a total of $2*10+5*5=45$ - (We can't spend less than 2 dimes and no quarters because we would have to spend more than 5 nickels) With these 2 ways to pay, the total number of ways is $2+3=5$. A person with five nickels, three dimes, and two quarters can make a $45$-cent purchase form the machine in $5$ ways.