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Basic Counting Rules

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Applying the Fundamental Counting Principle

A veteran's license plate consists of digit (0-9) followed by a capital letter (A-Z) followed by a second digit (0-9). How many possible license plate combinations are there?

Guidance

Sometimes we want to know how many different combinations can be made of a variety of items. The fundamental counting principle which states that the number of ways in which multiple events can occur can be determined by multiplying the number of possible outcomes for each event together. In other words, if events A , B , and C have 5, 3 and 4 possible outcomes respectively, then the possible combinations of outcomes is 5 \times 3 \times 4=60 .

The following examples will aid in developing an understanding of this concept and its application.

Example A

Sofia works in a clothing store. He has been given the task of setting up a mannequin with a skirt, a shirt and a pair of shoes from a display of coordinating skirts, shirts and shoes. Since they all coordinate she can pick any shirt, any skirt and any pair of shoes and the outfit will work. If there are 3 skirts, 5 shirts and 2 pairs of shoes, how many ways can she dress the mannequin?

Solution: Let’s use a tree to help us visualize the possibilities. If we start with Shirt A, we get the following possibilities for the remainder of the outfit:

So we could have the following 6 combinations with Shirt A:

Shirt A, Skirt A, Shoe A

Shirt A, Skirt A, Shoe B

Shirt A, Skirt B, Shoe A

Shirt A, Skirt B, Shoe B

Shirt A, Skirt C, Shoe A

Shirt A, Skirt C, Shoe B

Consider that there are four other shirts that will also have 6 combinations of skirts and shirts that will go with them. Now, there are 5 \times 6 total combinations which is 30 ways that Sofia could dress the mannequin.

Example B

Ralph is trying to purchase a new car. The salesperson tells him that there are 8 different possible interior colors, 5 exterior colors and 3 car models to choose from. How many different unique cars does he have to choose from?

Solution: Instead of making a tree diagram this time, let’s look at a more efficient method for determining the number of combinations. If we consider what happens in the tree diagram, the 8 different interior colors would each be matched with each of the 5 exterior colors and those combinations would then be linked to the 3 different models, we can see that:

8 \ \text{interior colors} \times 5 \ \text{exterior colors} \times \ \text{3 models}= 8 \times 5 \times 3=120 \ \text{combinations}

Example C

Monique is having a 5 course dinner in the dining room on a cruise. The menu consists of 2 appetizers, 3 soups, 2 salads, 4 entrees and 3 desserts. How many different meals could be configured if she chooses one of each course?

Solution: Following the method described in example B, we can multiply the number of chooses for each course together to determine the total combinations:

2 \times 3 \times 2 \times 4 \times 3=144 \ \text{unique} \ 5 \ \text{course meals} .

Intro Problem Revisit We can multiply the number of choices for each license plate position together to determine the total combinations:

10 \times 26 \times 10=2600 .

Therefore, there are 2600 possible license plate combinations.

Guided Practice

1. A coffee shop offers a special espresso deal. You choose one of three sizes, one of 5 flavored syrups and whole, nonfat or soy milk. How many drink combinations can be made?

2. Sarah goes to a local deli which offers a soup, salad and sandwich lunch. There are 3 soups, 3 salads and 6 sandwiches from which to choose. How many different lunches can be formed?

3. A design your own t-shirt website offers 5 sizes, 8 colors and 25 designs for their shirts. How many different t-shirts can be designed?

Answers

1. There are 3 sizes, 5 syrups and 3 kinds of milk from which to choose. So, 3 \times 5 \times 3=45 \ \text{drinks} .

2. 3 \ \text{soups} \times 3 \ \text{salads} \times 6 \ \text{sandwiches}=54 \ \text{lunch combos} .

3. 5 \ \text{sizes} \times 8 \ \text{colors} \times 25 \ \text{designs}=1000 \ \text{shirts} .

Vocabulary

The Fundamental Counting Principle
states that the number of ways in which multiple events can occur can be determined by multiplying the number of possible outcomes for each event together.

Explore More

Use the Fundamental Counting Principle to answer the following questions.

  1. A frozen yogurt shop has a half price Sunday Sundae special. Customers can get one of four flavors, one of three syrups and one of twelve toppings on their sundae. How many possible sundae combinations can be made?
  2. At a neighborhood restaurant wings are the specialty. The restaurant offers 3 sizes of wings, 4 levels of heat and ranch or blue cheese dipping sauce. How many different orders are possible?
  3. A noodle restaurant offers five types of noodles to choose from and each dish comes with a choice of one of four meats and six different sauces. How many combinations can be made?
  4. Charlie flips a coin and then rolls a die. How many different outcomes are possible?
  5. On a one week cruise, the ship stops in four ports. At each port there are six different excursions to choose from. If a passenger chooses one excursion at each port, how many different vacation experiences can be created?
  6. Samuel wants to know if he can go a whole month without wearing the exact same outfit twice. He has three pairs of pants, six shirts and two pairs of shoes. Can he make a unique outfit for each day of the month?
  7. A car dealership has four different models to choose from in six exterior colors. If there are three different interior colors to choose from, how many different vehicles can be designed?
  8. A burrito bar offers a lunch special burrito. Customers can choose a flour or corn tortilla; chicken, steak or carnitas; white or brown rice; black beans or pinto beans; cheese, guacamole, or sour cream; and one of four salsas for a special price. How many different burritos can be made?
  9. Maria rolls a die, spins a spinner with four numbers and then flips a coin. How many possible outcomes are there?
  10. A local restaurant offers a dinner special. Diners can choose one of six entrees, one of three appetizers and one of 3 desserts. How many different meals can be formed?

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