Question

In Exercises 5–36, express all probabilities as fractions.

Change for a Quarter How many different ways can you make change for a quarter? (Different arrangements of the same coins are not counted separately.)

Short answer

The number of ways in which a change for a quarter can be made is equal to 12.

Step-by-step solution

Given information

A quarter is to be formed using change.

Describe the counting rule

The number of possibilities in which an event can occur happen are counted according as per the underlying conditions of the experiment.

The most convenient method is tocount the entire set of combinations once they are listed.

List the possible combinations to obtain a quarter

The value of a quarter is $0.25.

Only three coins that are less than a quarter are a nickel ($0.05), a penny ($0.01) and a dime ($0.10), combination of which can be used to provide the value equivalent to a quarter.

The number of ways in which a quarter can be made is listed below:

• 2 dimes and 1 nickel as:

$2\left(0.10\right)+1\left(0.05\right)=0.25$

• 2 dimes and 5 pennies as:

$2\left(0.10\right)+5\left(0.01\right)=0.25$

• 1 dime and 3 nickels as:

$1\left(0.10\right)+3\left(0.05\right)=0.25$

• 1 dime, 2 nickels and 5 pennies as:

$1\left(0.10\right)+2\left(0.05\right)+5\left(0.01\right)=0.25$

• 1 dime, 1 nickel and 10 pennies as:

$1\left(0.10\right)+1\left(0.05\right)+10\left(0.01\right)=0.25$

• 1 dime and 15 pennies as:

$1\left(0.10\right)+15\left(0.01\right)=0.25$

• 5 nickels as:

$5\left(0.05\right)=0.25$

• 4 nickels and 5 pennies as:

$4\left(0.05\right)+5\left(0.01\right)=0.25$

• 3 nickels and 10 pennies as:

$3\left(0.05\right)+10\left(0.01\right)=0.25$

• 2 nickels and 15 pennies as:

$2\left(0.05\right)+15\left(0.01\right)=0.25$

• 1 nickel and 20 pennies as:

$1\left(0.05\right)+20\left(0.01\right)=0.25$

• 25 pennies as:

$25\left(0.01\right)=0.25$

Therefore, there are 12 ways to make a quarter using nickels, dimes and pennies.