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M ARIO F . T RIOLA

S TATISTICS. E LEMENTARY. Section 3-6 Counting. M ARIO F . T RIOLA. E IGHTH. E DITION. 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. T & a T & b T & c T & d T & e F & a

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M ARIO F . T RIOLA

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  1. STATISTICS ELEMENTARY Section 3-6 Counting MARIO F. TRIOLA EIGHTH EDITION

  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. T & a T & b T & c T & d T & e F & a F & b F & c F & d F & e a b c d e a b c d e T F Tree Diagram of the events 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. T & a T & b T & c T & d T & e F & a F & b F & c F & d F & e a b c d e a b c d e T F Tree Diagram of the events Let m represent the number of ways the first event can occur. Let n represent the number of ways the second event can occur. m = 2 n = 5 m*n = 10

  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. Example An ATM pin number is a 4 digit number. How many possible pin numbers are there, if you allow repeats in each position? Digit: 1st 2nd 3rd4th # of Choices: 10 10 10 10 By the FCR, the total number of possible outcomes are: 10 * 10 * 10 * 10 = 10,000

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

  8. Example 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 )

  9. Notation • 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 !

  10. Factorial Rule An entirecollection of ndifferent 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

  11. Example 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)!

  12. Permutations Rule • nis the number of available items (none identical to each other) • ris the number of items to be selected • the number of permutations (or sequences) is P n! n r = (n – r)! • Orderis taken into account

  13. n! n1! . n2! .. . . . . . . nk! Permutations Rule when some items are identical to others • If there are n items with n1 alike, n2 alike, . . . nk alike, the number of permutations is

  14. 11! =34,650 4!4!2! Permutations Rule • How many ways can the letters in MISSISSIPPI be arranged? I occurs 4 times S occurs 4 times P occurs 2 times

  15. Example 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 ofTop 3 with orderare8!/(8-3)! = 336 • For each chosen top three, if we rank/order them, there are 3! possibilities. ==> the number of choices ofTop 3 without order are {8!/(8-3)!}/(3!) = 56 8! (8-3)! 3! = Combinations rule

  16. Combinations Rule • the number of combinations is n! • n different items • r items to be selected • different orders of the same items are notcounted nCr= (n – r )! r!

  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: 10P5 = 10 nPr 5 = 30240 10C5 = 10 nCr 5 = 252

  18. Counting Devices Summary • 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.

  19. Counting Devices Summary • Are there n different items with all of them to be used in different arrangements? If so, use the factorial rule and findn!.

  20. Counting Devices Summary • Are there n different items with some of them to be used in different arrangements? If so, evaluate n! n Pr = (n – r )!

  21. Counting Devices Summary • 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 nitems? If so, use the following expression, in which n1 of the items are alike, n2 are alike and so on n! n1!n2!. . . . . . nk!

  22. Counting Devices Summary • 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 n! nCr = (n – r )! r!

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