Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Counting Section 3-7 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 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.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3 abcdeabcdeabcdeabcde TFTF Tree Diagram of the events T & a T & b T & c T & d T & e F & a F & b F & c F & d F & e

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 abcdeabcdeabcdeabcde TFTF Tree Diagram of the events 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

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 Rank three players (A, B, C). How many possible outcomes are there? Ranking : First Second Third Number of Choices : By FCR, the total number of possible outcomes are: 3 * 2 * 1 = 6 ( Notation: 3! = 3*2*1 ) Example

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6  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  Find the ! key on your calculator Notation

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 A collection of n different items can be arranged in order n ! different ways. Factorial Rule

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

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 9  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

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10  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!

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 when some items are identical to others Permutations Rule

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 12 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

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!

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14 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)!  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!) Example 8! (8-3)! 3! =

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)!  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!) Example 8! (8-3)! 3! = Combinations rule!

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16  the number of combinations is Combinations Rule

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17  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

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18 When different orderings of the same items are to be counted separately, we have a permutation problem, but when different orderings are not to be counted separately, we have a combination problem.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 19  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

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 20  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

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 21  Are there n different items with some of them to be used in different arrangements? Counting Devices Summary

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 22  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 =

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 23  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? Counting Devices Summary

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 24  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 !

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 25  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)? Counting Devices Summary

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 26  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!