1. 8.3 Counting Apply the fundamental counting principle
Calculate and apply permutations
Calculate and apply combinations

2. Tree Diagram (1 of 2)
Suppose that a quiz has only two questions. The first is a multiple-choice question with four choices, A, B, C, or D, and the second is a true-false (T-F) question. A tree diagram can be used to count the ways that this quiz can be answered.

3. Tree Diagram (2 of 2)
A tree diagram is a systematic way of listing every possibility. We can see that there are eight ways to answer the test. They are AT, AF, BT, BF, CT, CF, DT, and DF. For instance, CF indicates a quiz with answers of C on the first question and F on the second question.

4. Fundamental Counting Principle
Let E1, E2, E3,…, En be a sequence of n independent events. If event Ek can occur mk ways for k = 1, 2, 3,…, n, then there are m1 • m2 • m3 • … • mn ways for all n events to occur. Two events are independent if neither event influences the outcome of the other.

5. Example: Counting ways to answer an exam (1 of 2)
An exam contains four true-false questions and six multiple-choice questions. Each multiple-choice question has five possible answers. Count the number of ways that the exam can be answered. Solution This is a sequence of 10 independent events. There are two ways to answer each of the first four questions. There are five ways to answer the next 6 questions.

6. Example: Counting ways to answer an exam (2 of 2)
The number of ways to answer the exam is

7. Example: Counting license plates (1 of 2)
Sometimes a license plate is limited to 3 uppercase letters (A through Z) followed by 3 digits (0 through 9). For example, ABB 112 would be a valid license plate. Would this format provide enough license plates for a state with 8 million vehicles?

8. Example: Counting license plates (2 of 2)
Solution 26 choices for letters 10 choices for digits There are 26 ways to choose each of the three letters and 10 ways to choose each of the 3 digits.
This format for license plates could accommodate more than 8 million vehicles.

9. Example: Counting toll-free telephone numbers (1 of 2)
Count the total number of 800 numbers if the local portion of a telephone number (the last seven digits) does not start with a 0 or 1. Solution A toll-free 800 number assumes the following form.

10. Example: Counting toll-free telephone numbers (2 of 2)
We can think of choosing the remaining digits for the local number as seven independent events. Since the local number cannot begin with a 0 or 1, there are eight possibilities (2 to 9) for the first digit. The remaining six digits can be any number from 0 to 9, so there are ten possibilities for each of these digits. The total is given by

11. Permutations
A permutation is an ordering or arrangement. For example, suppose that three students are scheduled to give speeches in a class. The different arrangements in which these speeches can be ordered are called permutations.

12. n-Factorial

13. Example: Calculating factorials (1 of 2)

14. Example: Calculating factorials (2 of 2)

15. Permutations of n Elements Taken r at a Time
If P(n, r) denotes the number of permutations of n elements taken r at a time, with r ≤ n, then

16. Example: Calculating permutations
For a class of 30 students, how many arrangements are there in which 4 students each give a speech? Solution The number of permutations of 30 elements taken 4 at a time is given by
There are 657,720 ways to arrange the four speeches.

17. Combination (1 of 2)
A combination is not an ordering or arrangement, but rather a subset of a set of elements. Order is not important when finding combinations.

18. Combination (2 of 2)
For example, suppose we want to select a tennis team of two players from four people. The order in which the selection is made does not affect the final team of two players. From a set of four people, we select a subset of two players.

19. Combinations of n Elements Taken r at a Time
If C(n, r) denotes the number of combinations of n elements taken r at a time, with r ≤ n, then

20. Example: Calculating C(n, r) (1 of 2)
Calculate each of the following. Support your answer by using a calculator.
Solution

21. Example: Calculating C(n, r) (2 of 2)

22. Example: Counting combinations (1 of 2)
A college student has five courses left in her major and plans to take two of them this semester. Assuming that this student has the prerequisites for all five courses, determine how many ways these two courses can be selected.

23. Example: Counting combinations (2 of 2)
Solution The order in which the courses are selected is unimportant. From a set of 5 courses, the student selects a subset of 2 courses. The number of subsets is
There are 10 ways to select two courses from a set of five.

24. Example: Counting committees (1 of 2)
How many committees of six people can be selected from six women and three men, if a committee must consist of at least two men?
Solution
Two Men: Committee would be two men and four women.

25. Example: Counting committees (2 of 2)
Three Men: Committee would be three men and three women
The total number of possible committees would be = 65