1. Chapter 7 - Part Two Counting Techniques Wednesday, March 18, 2009

2. Permutations:  A permutation of r elements from a set of n elements in any specific ordering or arrangement, without repetition of the r elements. Each arrangement is a different permutation.  Clue words: arrangement, schedule, order,....  Example: There are six permutations of the letters A, B, and C.  ABC ACB BAC BCA CBA CAB

3. Permutations Formula

4. Example  In the Olympics Gymnastics competition, 8 gymnasts compete for medals. How many ways can the medals be awarded (gold, silver, and bronze)?

5. Distinguishable Permutations  Objects are not all distinguishable, namely n 1 of type 1, n 2 of type 2, etc. The number of permutations is:

6. Example  How many permutations are there of the letters in the word STATISTICS?

7. Combinations  A subset of items selected without regard to order.  Clue words: group, committee, sample....  Example: There is only one combination of the letters A, B, and C === ABC

8. Combinations Formula

9. Example  How many committees of three people can be selected from a group of 8 people?

10. Pascal’s Triangle  Can be used to compute combinations

11. Baseball  How many ways can three outfielders and four infielders be chosen from five outfielders and seven infielders?

12. Lottery  In the Pennsylvania lottery drawing, 5 numbered balls are selected from a box containing balls numbered 1 through 40. How many different combinations of winning numbers are there?

13. Some Card Problems  I am playing a hand of 5 card poker. What is the probability that I am dealt the following: 3 Kings and 2 Aces? All hearts Exactly two aces Three of a kind.

14. Example  A barrel contains 15 apples. Of the apples, 5 are rotten and 10 are good. Three apples are selected at random. What is the probability of selecting at least one good apple?

15. Binomial Probability  Same experiment is repeated several times.  Only two possible outcomes: success and failure  Repeated trials are independent.  n = number of trials  x = number of successes  p = probability of success on each trial

16. Formula

17. Example  Flip a coin 20 times. What is the probability of getting 6 tails?

18. Example  I am taking a 10 question, multiple choice exam and I have not studied. Each question has 4 possible answers. By guessing only, what is the probability that I can get 6 questions correct?

19. Problem of the Day  A customer walks into a hardware store to buy something and asks the clerk how much 1 would cost and the clerk answers $1. The customer then asks how much 10 would cost and the clerk answers $2. The customer says, “I’ll buy 1515,” and pays the clerk $4. What was the customer buying?