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Lesson 22. Fundamental Counting Principle and Permutations Advanced Algebra B Reference: McDougal Littell: §10.1 Apply Properties of Real Numbers. Pages 682-686. See Examples 1-6 The Fundamental Counting Principle: If you have two (or more) events with m and n choices respectively, the number of ways the BOTH events can occur is m·n. n-factorial is the product: n(n - 1)(n - 2)·...·3·2·1 Permutations. Distinguishable Permutations. The number of permutations (P) of n The number of distinguishable permutations of n objects if one elements taken r at a time is given by: object is repeated s1 times and another s2 times and so on, is given n! n! n Pr by: ( n r )! s1! s 2 !... Example 1. Counting Principle. (a) A customer in a computer store can choose 1 of 3 monitors, 1 of 2 keyboards and 1 of 4 printers. Assuming all components are compatible, how many different systems can be chosen? Make a “tree diagram” (b) At a used book sale, you are interested in 5 novels, 3 books of non-fiction, and 7 comic books. If you buy one of each kind, how many different choices do you have? (c) The digits 0, 1, 2, 3 and 4 are used to generate four-digit customer codes. How many different codes are possible: - if digits can be repeated? - if digits can’t be repeated? (d) In the state of New York, license plates consist of 3 letters followed by 4 numbers. How many different license plates are possible: - if letters and numbers can be repeated? - if letters and numbers can’t be repeated? Example 2. Ordering. (a) Write all of the different arrangements of the set of letters {s,b,c} (b) How many ways can you arrange 5 books on a shelf? (c) Seven sales girls are to be assigned to seven different counters in a department store. In how ways can the assignment be made? Example 3. Permutations. (a) How many ways can 5 books be arranged 3 at a time? (b) How many different ways can 4 raffle tickets be selected from a group of 50 tickets if each ticket wins a different prize? (c) A television news anchor has 8 different stories to present on the evening news: - How many different ways can the stories be presented? - If only three of the stories can be presented, how many ways can a lead story, a second story and a closing story be presented? Example 4. Distinguishable Permutations. (a) Write all of the possible permutations of the word BOB. (b) Find the number of distinguishable permutations of the letters in MIAMI (c) Find the number of distinguishable permutations of the letters in TALLAHASSEE. Additional Problems. 1. p. 686 #3 2. p. 686 #8 3. p. 686 #14 4. p. 686 #16 5. p. 686 #18 6. p. 686 #25 7. p. 686 #26 8. p. 686 #28 9. p. 686 #29 10. p. 686 #30 11. p. 686 #36 12. p. 686 #44 13. p. 686 #52 14. p. 686 #61 15. p. 686 #63 16. p. 686 #65 17. p. 686 #71 18. Three couples have reserved seats in a given row at a concert. How many different ways can they be seated if: (a) there are no seating restrictions? (b) the couples wish to sit together? (c) the boys wish to sit together? 19. In how many ways can eight students be seated in a row if two particular students must not be seated next to each other? 20. How many different 3 letter “words” can be made from the letters in MATRICES? Review Simplify. 2 3 2x 5 x 4 24. x x 1 1 3 x 5 21. 22. 2 23. x x2 x 4x 3 x 3 3x 1 3x 1 1 2x 2