1. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 3-6 Counting

2. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 Assume a quiz consists of two questions. A true/false and a multiple choice with 5 possible answers. How many different ways can they occur together.

3. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3 Tree Diagram of the events abcdeabcdeabcdeabcde TFTF T & a T & b T & c T & d T & e F & a F & b F & c F & d F & e Assume a quiz consists of two questions. A true/false and a multiple choice with 5 possible answers. How many different ways can they occur together.

4. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 Tree Diagram of the events abcdeabcdeabcdeabcde TFTF T & a T & b T & c T & d T & e F & a F & b F & c F & d F & e m = 2 n = 5 m*n = 10 Let m represent the number of ways the first event can occur. Let n represent the number of ways the second event can occur.

5. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 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, together the events can occur a total of m n ways.

6. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 An ATM pin number is a 4 digit number. How many possible pin numbers are there, if you allow repeats in each position? Digit : 1 st 2 nd 3 rd 4 th # of Choices : 10 10 10 10 By the FCR, the total number of possible outcomes are: 10 * 10 * 10 * 10 = 10,000 Example

7. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 An ATM pin number is made up of 4 digit number. How many possible outcomes are there, if no repeats are allowed? Digit :1 st 2 nd 3 rd 4 th # of Choices : 10 9 8 7 By the FCR, the total number of possible outcomes are: 10 * 9 * 8 * 7 = 5040 Example

8. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8 Rank three players (A, B, C). How many possible outcomes are there? Ranking : First Second Third Number of Choices : 3 2 1 By FCR, the total number of possible outcomes are: 3 * 2 * 1 = 6 ( Notation: 3! = 3*2*1 ) Example

9. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 9  The factorial symbol ! denotes the product of decreasing positive whole numbers.  n ! = n (n – 1) (n – 2) (n – 3) (3) (2) (1)  Special Definition: 0 ! = 1  The ! key on your TI-8x calculator is found by pressing MATH and selecting PRB and selecting choice #4 ! Notation

10. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10 An entire collection of n different items can be arranged in order n ! different ways. Example: How many different seating charts could be made for a class of 13? 13! = 6,227,020,800 Factorial Rule

11. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 Eight players are in a competition, three of them will win prizes (gold/silver/bronze). How many possible outcomes are there? Prizes : gold silver bronze Number of Choices : 8 7 6 By FCR, the total number of possible outcomes are: 8 * 7 * 6 = 336 = 8! / 5! = 8!/(8-3)! Example

12. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 12  n is the number of available items (none identical to each other)  r is the number of items to be selected  the number of permutations (or sequences) is Permutations Rule  Order is taken into account P n r = ( n – r ) ! n!n!

13. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 13 when some items are identical to others  If there are n items with n 1 alike, n 2 alike,... n k alike, the number of permutations is Permutations Rule n 1 !. n 2 !........ n k ! n!n!

14. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14  How many ways can the letters in MISSISSIPPI be arranged? I occurs 4 times S occurs 4 times P occurs 2 times Permutations Rule 4 ! 4 ! 2 ! 11 ! =34,650

15. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15 Eight players are in a competition, top three will be selected for the next round (order does not matter). How many possible choices are there?  By the Permutations Rule, the number of choices of Top 3 with order are 8!/(8-3)! = 336  For each chosen top three, if we rank/order them, there are 3! possibilities. ==> the number of choices of Top 3 without order are { 8!/(8-3)!}/(3!) = 56 Example 8! (8-3)! 3! = Combinations rule

16. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16  n different items  r items to be selected  different orders of the same items are not counted  the number of combinations is (n – r )! r! n!n! n C r = Combinations Rule

17. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17 Permutation –Order Matters Combination - Order does not matter TI-83/4 Press MATH choose PRB choose 2: nPr or 3: nCr to compute the # of outcomes. Example: 10 P 5 = 10 nPr 5 = 30240 10 C 5 = 10 nCr 5 = 252

18. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18  Is there a sequence of events in which the first can occur m ways, the second can occur n ways, and so on? If so use the fundamental counting rule and multiply m, n, and so on. Counting Devices Summary

19. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 19  Are there n different items with all of them to be used in different arrangements? If so, use the factorial rule and find n!. Counting Devices Summary

20. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 20  Are there n different items with some of them to be used in different arrangements? If so, evaluate Counting Devices Summary (n – r )! n!n! n P r =

21. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 21  Are there n items with some of them identical to each other, and there is a need to find the total number of different arrangements of all of those n items? If so, use the following expression, in which n 1 of the items are alike, n 2 are alike and so on Counting Devices Summary n!n! n 1 ! n 2 !...... n k !

22. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 22  Are there n different items with some of them to be selected, and is there a need to find the total number of combinations (that is, is the order irrelevant)? If so, evaluate Counting Devices Summary n!n! n C r = (n – r )! r!