# 4.2: Counting

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We encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. We will examine this type of mathematics in this section.

### Tree Diagram

There are several methods to count the number of outcomes. Below is a tree diagram example where each option branches to each of the next options.

##### Example $$\PageIndex{1}$$

Suppose we are choosing an appetizer, an entrée, and a dessert. The options for an appetizer is soup or salad, the options for an entrée are chicken, fish, or steak, and the options for dessert are cake or pudding. These options are on a fixed-price dinner menu; create a tree diagram to count the number of total options.

Solution

The tree diagram shows there are 12 dinner options.

### Listing Method

The listing method just lists each possible outcome.

##### Example $$\PageIndex{2}$$

Suppose we are choosing an appetizer, an entrée, and a dessert. The options for an appetizer is soup or salad, the options for an entrée are chicken, fish, or steak, and the options for dessert are cake or pudding. These options are on a fixed-price dinner menu; list all the options to count the number of total options.

Solution

It is recommended to list all the options in a methodical manner to help make sure an option isn't missed. So list all the options where soup is the appetizer first, then go through the entrée options in order, with each dessert.

1. soup, chicken, cake
2. soup, chicken, pudding
3. soup, fish, cake
4. soup, fish, pudding
5. soup, steak, cake
6. soup, steak, pudding

There are 12 options.

### The Fundamental Counting Principle

Both the previous methods can be lengthy, time-consuming, and it may be easy to miss an option resulting in an incorrect answer. A faster method may be to use the Fundamental Counting Principle.

##### Definition: Fundamental Counting Principle

The Fundamental Counting Principle says, if one event can occur in $$m$$ ways and a second event can occur in $$n$$ ways after the first event has occurred, then the two events can occur in $$m \times n$$ ways.

##### Example $$\PageIndex{3}$$

Suppose we are choosing an appetizer, an entrée, and a dessert. The options for an appetizer is soup or salad, the options for an entrée are chicken, fish, or steak, and the options for dessert are cake or pudding. These options are on a fixed-price dinner menu; use the Fundamental Counting Principle to count the number of total options.

Solution

$\text{number of appetizer options}\times \text{number of entree options} \times \text{number of dessert options}$

$2 \times 3 \times 2=12$

There are 12 options for dinner.

Using the Fundamental Counting Principle can save a lot of time.

##### Example $$\PageIndex{3}$$

Diane packed 2 skirts, 4 blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for each outfit and decide whether to wear the sweater. Use the Fundamental Counting Principle to find the total number of possible outfits.

Solution

To find the total number of outfits, find the product of the number of skirt options, the number of blouse options, and the number of sweater options.

$\text{number of skirt options}\times \text{number of blouse options} \times \text{number of sweater options}$

$2 \times 4 \times 2=16$

There are 16 possible outfits.

##### Exercise $$\PageIndex{1}$$

A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are 3 types of breakfast sandwiches, 4 side dish options, and 5 beverage choices. Find the total number of possible breakfast specials.