The Fundamental Counting Principle Section 2 Objectives :

1. The Fundamental Counting PrincipleSection 2 Objectives: • Understand the fundamental counting principle. • Use slot diagrams to organize information in counting problems. • Know how to solve counting problems with special conditions.

2. The Fundamental Counting Principle • How many ways can you flip a coin, roll one die and then flip another coin? • Draw a tree diagram to list all possible outcomes. • How many ways can you put on your pants and a shirt if you have two pairs of pants and three shirts? • Draw a tree diagram to list all possible outcomes.

3. The Fundamental Counting Principle cont. (2) A group is planning a fund-raising campaign featuring two endangered species, one animal for TV commercials and one for use online. The list of candidates includes the (C)heetah, the (O)tter, the black-footed (F)erret, and the Bengal (T)iger. In how many ways can we choose the two animals for the campaign? Solution: We first choose an animal for the TV campaign, which can be done in four ways. We then choose a different animal for the online ads, which can be done in three ways. So the total number of ways to choose the animals is

4. The Fundamental Counting Principle cont. (3) • How many ways can four coins be flipped? Solution: Flipping the first coin can be done in two ways. Flipping the second, third, and fourth coins can also each be done in one of two ways. The four coins can be flipped in • How many ways can three dice (red, green, blue) be rolled? • Solution: Rolling the red die can be done in six ways. • Rolling the green and blue dice can also each be done in one • of six ways. The three dice can be rolled in

5. The Fundamental Counting Principle cont. (4) A summer intern wants to vary his outfit by wearing different combinations of coats, pants, shirts, and ties. If he has three sports coats, five pairs of pants, seven shirts, and four ties, how many different ways can he select an outfit consisting of a coat, pants, shirt, and tie? Solution: The interns options are

6. Slot Diagrams A useful technique for solving problems involving various tasks is to draw a series of blank spaces to keep track of the number of ways to do each task. We will call such a figure a slot diagram. ___ ___ ___ ___ ___

7. Slot Diagrams cont. (2) A security keypad uses five digits (0 to 9) in a specific order. How many different keypad patterns are possible if any digit can be used in any position and repetition is allowed? Solution: The slot diagram indicates there are 10 × 10 × 10 × 10 × 10 = 100,000 possibilities. ___ ___ ___ ___ ___

8. Slot Diagrams cont. (3) In the previous example, suppose the digit 0 cannot be used as the first digit, but otherwise any digit can be used in any position and repetition is allowed. Solution: The slot diagram indicates there are 9 × 10 × 10 × 10 × 10 = 90,000 possibilities.

9. Slot Diagrams cont. (4) In the previous example, suppose any digit can be used in any position, but repetition is not allowed? Solution: The slot diagram indicates there are 10 × 9 × 8 × 7 × 6 = 30,240 possibilities.

10. Handling Special Conditions A college class has 10 students. Louise must sit in the front row next to her tutor. If there are six chairs in the first row of the classroom, how many different ways can students be assigned to sit in the first row? Solution: We first consider the following tasks • Task 1: Assign two seats to Louise and her tutor. • Task 2: Arrange Louise and her tutor in these two seats. • Task 3: Assign the remaining seats.

11. Handling Special Conditions cont. (2) • Task 1: There are five ways to assign 2 seats to Louise and her tutor. • Task 2: There are two ways that Louise and her tutor can sit in their seats—Louise sits either on the right or the left.

12. Handling Special Conditions cont. (2) • Task 3: The remaining four seats are to be filled by four of the eight students left; thus, we have eight students for the first remaining seat, seven for the second seat, and so on. So there are 16,800 ways that students can be assigned to sit in the first row.