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Unit 9 Counting Principles

Unit 9 Counting Principles. Counting Outcomes! Discovery in Groups. Counting Outcomes! Discovery in Groups. Counting Outcomes! Discovery in Groups. Counting Outcomes! Discovery in Groups Show one step of work!. Counting Outcomes!. Counting Outcomes! Discovery in Groups. Finish.

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Unit 9 Counting Principles

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  1. Unit 9Counting Principles

  2. Counting Outcomes!Discovery in Groups

  3. Counting Outcomes!Discovery in Groups

  4. Counting Outcomes!Discovery in Groups

  5. Counting Outcomes!Discovery in GroupsShow one step of work!

  6. Counting Outcomes!

  7. Counting Outcomes!Discovery in Groups • Finish

  8. Lesson A – Multiplication Property and PermutationsHow many different basic license plate combinations are there available in the state of Iowa? Or the number of combo meals you can choose from at your favorite fast food place

  9. FUNDAMENTAL COUNTING PRINCIPLE: • Suppose you have n choices to make, with m1 ways to make choice 1, m2 ways to make choice 2, m3 ways to make choice 3, etc., and mn choices to make choice n. Then there are • different ways

  10. Examples: • a) A standard state of Iowa license plate included three letter characters and three number characters: How many different basic license plate combinations are possible? • b)And if there are 10 different sandwiches, 5 different side items, and 8 different drinks to choose from at Wendy’s, then how many different combo meals are available if you must choose one of each?

  11. c) A lock can be set to open to a 3-character sequence using letters and/or numbers • (0-9). How many possible lock combinations are available? • d) How many combinations are available if each character must be unique (no repeats) in the sequence.

  12. Skip • E) Challenge

  13. Phone break

  14. Phone break • Then use your phone to determine if 0 is even or odd or neither and be ready • to share with the class.

  15. Try It: Complete on bottom of middle of page 2 of notes A lock can be set to a 4 digit passcode using #0-9. • How many 4-digit passcode are possible if the first digit must be even? • If you can not repeat any numbers? • If you have to have the last two numbers odd and cannot be repeated (on the last two)? • You have the second number as 4 and the third number as even?

  16. The use of the multiplication principle when order matter often leads to products such as 5 * 4 * 3 * 2 * 1. Another way to express a product of all natural numbers from n down to 1 is n! (reads n factorial). • Factorials • n! = n * n – 1 * n – 2 * … 1 5! = 5 * 4 * 3 * 2 * 1

  17. Examples: • How many ways can 8 books be stacked? Find key on Calculator

  18. How many ways can 3 of the 8 books • be stacked?

  19. The last example is called a permutation of 8 things taking 3 at a time. In these types of problems order matters and r elements are being picked out of n total elements available. Each re-arrangement of elements picked is a different permutation. • Permutations: If P(n, r) is the number of permutations of n elements taking r at a time, then • Another notation for P(n, r) is nPr

  20. Examples: 1) In a class of 25 students, 4 students will be picked as volunteers for an activity. Each student will be given a different role (recorder, timer, writer, and performer). How many ways can this group be formed?

  21. Examples: • 2) How many ways can a 5 of 18 players on a soccer team be selected for a shootout at different shooting spots. •  3) A gameshow selects 8 people from a crowed of 60, and gives each person selected a different prize. How many ways can be prizes be given out?

  22. HW Packet Page 1 #1-8, 13-16 Page 3 #43 MML Unit 9- 8.1- 12 questions- Due Monday

  23. HW Packet Page 1 #1-8, 13-16; Pg. 3 #43 • 720 2) 5040 3) 1.308 X 1012 • 2.092 X 1013 5) 156 6) 1320 • 1.024 X 1025 8) 9.960 X 1025

  24. HW Packet Page 1 #1-8, 13-16; Pg. 3 #43 • 720 2) 5040 3) 1.308 X 1012 • 2.092 X 1013 5) 156 6) 1320 • 1.024 X 1025 8) 9.960 X 1025 • 36 14) 168 15) 20 16) 3360 43) a) 27,600 b) 35,152 c) 1104

  25. Pg. 359 #3-9 (odds) #13-16, 21, 23, 25, 34, 37, 39, 40, 43

  26. Pg. 359 #3-9 (odds) #13-16, 21, 23, 25, 34, 37, 39, 40, 43 Check odds in back of book 14) 168 16) 3360 34) 95,040 40) 1,256,640

  27. DISTINGUISHABLE and NON – DISTIGUISHABLE PERMUTATIONS Pg. 5 of notes • Consider the following… • How many ways can the seven letters from the word “article” be arrange? • Answer: 7 * 6 * 5 * 4 * 3 * 2 * 1 = 7! = 5040 ways • In this example, each letter is distinguishable or non repeating.

  28. Consider this example… • How many ways can the letters from the work “Mississippi” be arranged? • This time, some letter repeat.

  29. Need to add ! On bottom part of equation • If n objects in a permutation are not all distinguishable – that is there are n1 of type 1, n2 of time 2, and so on for r types, then the number of distinguishable permutations is...

  30. 10 seniors, 5 Juniors, 5 sophomores, and 3 teachers are on a committee. For a photo each subgroup (seniors, juniors, sophomores, and teachers) will wear a different color shirt. How many ways can they be lined up for the photo according to color?

  31. Pg. 6 of notes • How many ways can the letters of the word Tennessee be arranged?

  32. Try It Find the number of unique permutations of the letters in each word. • MATH • BILLIONAIRE

  33. Try It Find the number of unique permutations of the letters in each word. • MATH 4! = 24 • BILLIONAIRE

  34. REVIEW WKST 8.1

  35. Wkst: Practice 8.1 • 1) A student buys 3 cherry yogurts, 2 raspberry yogurts, and 2 blueberry yogurts. She puts them in her dormitory refrigerator to eat one a day for the next week. Assuming yogurts of the same flavor are indistinguishable, in how many ways can she select yogurts to eat for the next week?

  36. Wkst: Practice 8.1 • 2) If we have six people for a picture, and you choose two, how many different ways can they be arranged? • 3) How many ways can you arrange the letters in the word PARALLEL

  37. WKST PRACTICE 8.1 • 210 • 30 • 3360

  38. Wkst Practice/Review 8.1 • #44) • A) 78,125 • B) 900,000 • C) 90,000 • D) 10,000 • E) 544,320 • #45) • 160; 8,000,000

  39. Pg. 4 Of Notes Lesson B – More on Multiplication and Permutations • Part a-e in groups • Work together then have me check • Hints: • A) 5000 < x < 5100 • B) 800 < x< 900 • C) 140 < x < 150 • D) 280 < x < 290 • E) 10 < x < 15

  40. Pg. 4 Of Notes Lesson B – More on Multiplication and Permutations • A talk show will have a panel of 7 guests, 3 men and 4 women, on the next episode. How many ways can the 7 guests be seated on the set? b) If you choose 4 of 7 guests to be seated on the set, how many ways can they be ordered?

  41. c) How many ways can the guests be seated if men and women are to be alternated? • d) How many ways can the guests be seated in all the women must sit together, and all the men must sit together? • e) How many ways can 1 man and one women be selected from the panel?

  42. On back of wkst:Pg. 359-360 • #21, 23, 25, 31, 33 • Complete and check in back of book

  43. MML work time- last 30 minutes • MML Unit 9 (8.1) Multiplication Principle & Permutations • Due Thursday at 11:50 pm • 12 problems

  44. Review permutations from yesterday • In your groups come up with an example that uses a distinguishable permutation and a non-distinguishable permutation and then solve

  45. Finish from yesterdayPg. 359-360 • #21, 23, 25, 31, 33, 45 • Pg. 346 • #52-58 • 52) 1/26 54) 8/13 56) 1/3 58) 0

  46. Practice/Review 8.1 #1-3

  47. Finish from yesterday • Pg. 359-360 • #21, 23, 25, 31, 33 • Pg. 359-360 #22, 32, 47, 48

  48. Finish from yesterday • Pg. 359-360 • #21, 23, 25, 31, 33 • Pg. 359-360 #22, 32, 47, 48 • 22) 540,540 32) 15,120

  49. Finish from yesterday • Pg. 359-360 • #21, 23, 25, 31, 33 • Pg. 359-360 #22, 32, 47, 48 • 22) 540,540 32) 15,120 48) 1,000,000,000; yes

  50. HW Packet Pg. 1 • a) 840 b) 180 c) 420 • 540,540 • A) 9! Or 362,880 or 9!/(4!3!2!) = 1260 b) 6 X 4!3!2! = 1728 c) 1260 d) 24 e) 144

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