1. Slide Slide 1 Created by Tom Wegleitner, Centreville, Virginia Edited by Olga Pilipets, San Diego, California Counting

2. Slide Slide 2 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Counting Techniques: Example 1 Procedure: you are flipping a coin 3 times in a row. a) Write a sample space for this procedure (use a tree diagram) b) What is the probability of getting exactly two heads? c) What is the probability that you get more than one tail?

3. Slide Slide 3 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Counting Techniques: Example 2 Ramona has designed 4 tops, 2 jackets, and 3 pairs of pants for her fashion show. How many different outfits can she create? Construct a tree diagram to answer the question.

4. Slide Slide 4 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Tree Diagrams A tree diagram is a picture of the possible outcomes of a procedure, shown as line segments emanating from one starting point. These diagrams are helpful if the number of possibilities is not too large. This figure summarizes the possible outcomes for a true/false followed by a multiple choice question. Note that there are 10 possible combinations.

5. Slide Slide 5 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Concept In many probability problems, the big obstacle is finding the total number of outcomes, and this section presents several methods for finding such numbers without directly listing and counting the possibilities.

6. Slide Slide 6 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Fundamental Counting Rule For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events together can occur a total of m n ways.

7. Slide Slide 7 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Fundamental Counting Rule: Example 3 #8 p.165. A new computer owner creates a password consisting of two characters. She randomly selects a letter of the alphabet for the first character and a digit for the second character. How many different passwords can be created? What is the probability that her password is “K9”?

8. Slide Slide 8 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Fundamental Counting Rule: Example 4 How many license plates could be created that have a digit, followed by two letters, followed by 3 digits?

9. Slide Slide 9 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Notation The factorial symbol ! denotes the product of decreasing positive whole numbers. For example, By special definition, 0! = 1.

10. Slide Slide 10 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 5 The word PARSIMONY has 9 letters. How many ways can we arrange letters from this word?

11. Slide Slide 11 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. A collection of n different items can be arranged in order n! different ways. (This factorial rule reflects the fact that the first item may be selected in n different ways, the second item may be selected in n – 1 ways, and so on.) Factorial Rule

12. Slide Slide 12 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 6 What is ?

13. Slide Slide 13 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 7 If there are 30 people in the classroom, how many ways can we select a president, vice president and a secretary? In this case we have three slots to fill, order is important, and there is no replacement.

14. Slide Slide 14 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Permutations Rule (ordering of distinct objects in a straight line.) (n - r)! n r P = n!n! If the preceding requirements are satisfied, the number of permutations (or sequences) of r items selected from n available items (without replacement) is Requirements: 1.There are n different items available. (This rule does not apply if some of the items are identical to others.) 2.We select r of the n items (without replacement). 3.We consider rearrangements of the same items to be different sequences. (The permutation of ABC is different from CBA and is counted separately.)

15. Slide Slide 15 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 8 A twelve-person community theatre group produces a yearly musical. How many ways can club members select one person from the group to direct the play, a second person to supervise the music, and the third person to handle publicity? Note: the order in which selection is made is important.

16. Slide Slide 16 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 9 The word MISSISSIPPI has 11 letters. How many ways can we arrange letters from this word? Note that some of the letters in this word are repeating.

17. Slide Slide 17 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Permutations Rule (when some items are identical to others) n 1 !. n 2 !........ n k ! n!n! If the preceding requirements are satisfied, and if there are n 1 alike, n 2 alike,... n k alike, the number of permutations (or sequences) of all items selected without replacement is Requirements: 1.There are n items available, and some items are identical to others. 2.We select all of the n items (without replacement). 3.We consider rearrangements of distinct items to be different sequences.

18. Slide Slide 18 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 9 The word MISSISSIPPI has 11 letters. How many ways can we arrange letters from this word? Note that some of the letters in this word are repeating.

19. Slide Slide 19 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 10 A twelve-person community theatre group produces a yearly musical. This time instead of assigning a specific duty to each person selected, the club members want to select a three-person committee to direct the play, to supervise the music, and to handle publicity. How many different committees of three people can be formed? Note: the order in which selection is made is not important. John-Mary-Chris is the same as Chris-John-Mary, and so on.

20. Slide Slide 20 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. (n - r )! r! n!n! n C r = Combinations Rule If the preceding requirements are satisfied, the number of combinations of r items selected from n different items is Requirements: 1.There are n different items available. 2.We select r of the n items (without replacement). 3.We consider rearrangements of the same items to be the same. (The combination of ABC is the same as CBA.)

21. Slide Slide 21 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 10 A twelve-person community theatre group produces a yearly musical. This time instead of assigning a specific duty to each person selected, the club members want to select a three-person committee to direct the play, to supervise the music, and to handle publicity. How many different committees of three people can be formed? Note: the order in which selection is made is not important. John-Mary-Chris is the same as Chris- John-Mary, and so on.

22. Slide Slide 22 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. 1.Selection is made without replacement 2.When the order in which the objects are selected is important we say we have a permutation of the objects. 3.When we are only interested in which objects were selected, not in the order in which they were selected, we are dealing with combinations. Permutations versus Combinations

23. Slide Slide 23 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Recap In this section we have discussed:  The fundamental counting rule.  The permutations rule (when items are all different).  The permutations rule (when some items are identical to others).  The combinations rule.  The factorial rule.