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Counting

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  1. Counting Just How Many Are There?

  2. The Fundamental Counting Principle 12.2 • Understand the fundamental counting principle. • Use slot diagrams to organize information in counting problems. • Know how to solve counting problems with special conditions.

  3. The Fundamental Counting Principle

  4. The Fundamental Counting Principle • Example: 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? (continued on next slide)

  5. The Fundamental Counting Principle • 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

  6. The Fundamental Counting Principle • Example: How many ways can four coins be flipped? • Solution:

  7. The Fundamental Counting Principle • Example: 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

  8. The Fundamental Counting Principle • Example: How many ways can three dice (red, green, blue) be rolled? • Solution:

  9. The Fundamental Counting Principle • Example: 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

  10. The Fundamental Counting Principle • Example: 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? (continued on next slide)

  11. The Fundamental Counting Principle • Solution: The interns options are

  12. 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.

  13. Slot Diagrams • Example: 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:

  14. Slot Diagrams • Example: 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.

  15. Slot Diagrams • Example: 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:

  16. Slot Diagrams • Example: 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.

  17. Slot Diagrams • Example: In the previous example, suppose any digit can be used in any position, but repetition is not allowed? • Solution:

  18. Slot Diagrams • Example: 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.

  19. Handling Special Conditions • Example: 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? (continued on next slide)

  20. Handling Special Conditions • Example: 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. (continued on next slide)

  21. Handling Special Conditions Task 1: There are five ways to assign 2 seats to Louise and her tutor. (continued on next slide)

  22. Handling Special Conditions 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. 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. total number possibilities