The Fundamental Counting Principle Section 12.2 The Fundamental Counting Principle

Objective: Use the Fundamental Counting Principle (FCP) to determine the number of possible outcomes in a given situation.

Key Term: Fundamental Counting Principle (FCP) – the number of ways in which a series of successive things can occur is found by multiplying the number of ways in which each thing can occur.

Example 1: Applying the Fundamental Counting Principle. The Greasy Spoon Restaurant offers 6 appetizers and 14 main courses. In how many ways can a person order a two-course meal?

Example 2: You are registering for college courses, there are 15 sections of psychology from which you can choose. Furthermore, there are 9 courses of social science that are available at times that do not conflict with those for psychology. In how many ways can you create two-course schedules that satisfy the psychological-social science requirement?

Example 3: Counting Outfits It’s early morning, you’re groggy, and you have to select something to wear for your 8 a.m. class. (What were you thinking when you signed up for a class at that hour?!) Fortunately your wardrobe is limited – just two pairs of jeans, three T-shirts, and two pair of sneakers to choose from. Use the FCP, how many outfits do you have to choose from?

Example 4: Using the FCP in Coin & Dice Problems How many ways can four coins be flipped? How many ways can three dice be rolled?

Example 5: Car of the Future Car manufacturers are now experimenting with lightweight three-wheel cars, designed for 1 person, and considered ideal for city driving. Intrigued? Suppose you could order such a car with a choice of 9 possible colors, with or without air conditioning, gas or electric powered, and with or without an onboard computer. In how many ways can this car be ordered with regard to these options?

Slot Diagrams Sometimes special conditions affect the number of ways we can perform the various tasks. A useful technique for solving problems such as these is to draw a series of blank spaces to keep track of the number of ways to do each task, such a figure is called a slot diagram.

Example 6: A Multiple-Choice Test You are taking a multiple-choice test that has ten questions. Each of the questions has four answer choices, with one correct answer per question. If you select one of these four choices for each question and leave nothing blank, in how many ways can you answer the question?

Example 7: Telephone Numbers Telephone numbers in the U. S. begin with three-digit area codes followed by seven-digit local telephone numbers. Area codes and local telephone numbers cannot begin with 0 or 1. How many different telephone numbers are possible?

Example 8: Electronic Gate An electronic gate can be opened by entering five digits on a keypad containing the digits 0, 1, 2,…,8, 9. How many different keypad sequences are possible if the digit 0 cannot be used as the first digit?

Example 9: Counting with Special Conditions Professor Frantz teaches an advanced psychology class of 10 students. She has a visually handicapped student, Louise, who must sit in the front row next to her tutor, who is also a member of the class. If there are six chairs in the first row of her classroom, how many different ways can Professor Frantz assign students to sit in the first row?

Example 10: Assigning responsibilities The board of an Internet startup company has seven members. If one person is to be in charge of marketing and another in charge of research, in how many ways can these two positions be filled?

Section 12.2 Assignment Classwork: TB pg. 692/2 – 26 Even and 32 – 36 Even Remember you must write problems and show ALL work to receive credit for this assignment.