Counting Principles: Learn It 1

  • Solve counting problems using the Addition and Multiplication Principle
  • Solve counting problems using permutations involving [latex]n[/latex] distinct objects
  • Solve counting problems using combinations
  • Find the number of subsets of a given set
  • Solve counting problems using permutations involving n non-distinct objects

A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. The customer can choose the order of the images and the letters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. Counting the possibilities is challenging!

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.

Using the Addition Principle

The company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet models and 5 supported smartphone models. The Addition Principle tells us that we can add the number of tablet options to the number of smartphone options to find the total number of options. By the Addition Principle there are 8 total options.

The addition of 3 iPods and 4 iPhones.

addition principle

The Addition Principle states that if one event can occur in [latex]A[/latex] ways ([latex]A[/latex] outcomes) and a second event can occur in [latex]B[/latex] ways ([latex]B[/latex] outcomes) and both events cannot occur at the same time ([latex]A[/latex] and [latex]B[/latex] disjoints), then there are [latex]A B[/latex] ways ([latex]A B[/latex] outcomes) for the first event OR the second event to occur.

A student is shopping for a new computer. He is deciding among [latex]3[/latex] desktop computers and [latex]4[/latex] laptop computers. What is the total number of computer options?

Using the Multiplication Principle

The Multiplication Principle applies when we are making more than one selection.

Suppose we are choosing an appetizer, an entrée, and a dessert. If there are [latex]2[/latex] appetizer options, [latex]3[/latex] entrée options, and [latex]2[/latex] dessert options on a fixed-price dinner menu, there are a total of [latex]12[/latex] possible choices of one each as shown in the tree diagram.A tree diagram of the different menu combinations.

  1. soup, chicken, cake
  2. soup, chicken, pudding
  3. soup, fish, cake
  4. soup, fish, pudding
  5. soup, steak, cake
  6. soup, steak, pudding
  7. salad, chicken, cake
  8. salad, chicken, pudding
  9. salad, fish, cake
  10. salad, fish, pudding
  11. salad, steak, cake
  12. salad, steak, pudding

We can also find the total number of possible dinners by multiplying.

Thus, there are [latex]12[/latex] possible dinner choices simply by applying the Multiplication Principle.

multiplication principle

The Multiplication Principle states that if one event can occur in [latex]A[/latex] ways ([latex]A[/latex] outcomes) and a second event can occur in [latex]B[/latex] ways ([latex]B[/latex] outcomes) after the first event has occurred then the two events can occur in [latex]A \cdot B[/latex] ways.

 

This is also known as the Fundamental Counting Principle.

Diane packed [latex]2[/latex] skirts, [latex]4[/latex] 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 Multiplication Principle to find the total number of possible outfits.