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# Integrated Math 2 Section 4-7

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Combinations of a Set

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• ### Integrated Math 2 Section 4-7

1. 1. SECTION 4-7 Combinations of a Set
2. 2. ESSENTIAL QUESTION • How do you ﬁnd the number of combinations of a set? • Where you’ll see this: • Cooking, travel, music, sports, games
3. 3. VOCABULARY 1. Combination: 2. nCr :
4. 4. VOCABULARY 1. Combination: The number of ways you can pick from a set of items when order is not important. 2. nCr :
5. 5. VOCABULARY 1. Combination: The number of ways you can pick from a set of items when order is not important. 2. nCr : The possible combinations where n is the total number of items and r is the number of items taken at a time
6. 6. VOCABULARY 1. Combination: The number of ways you can pick from a set of items when order is not important. 2. nCr : The possible combinations where n is the total number of items and r is the number of items taken at a time n! n Cr = (n − r )! r !
7. 7. PERMUTATIONS VS. COMBINATIONS If order is important: If order is not important:
8. 8. PERMUTATIONS VS. COMBINATIONS If order is important: Permutation If order is not important:
9. 9. PERMUTATIONS VS. COMBINATIONS If order is important: Permutation If order is not important: Combination
10. 10. PERMUTATIONS VS. COMBINATIONS If order is important: Permutation If order is not important: Combination n! n Pr = (n − r )!
11. 11. PERMUTATIONS VS. COMBINATIONS If order is important: Permutation If order is not important: Combination n! n! n Pr = n Cr = (n − r )! (n − r )! r !
12. 12. EXAMPLE 1 How many ways can you select 3 committee members from a group of 5 people?
13. 13. EXAMPLE 1 How many ways can you select 3 committee members from a group of 5 people? Is order important?
14. 14. EXAMPLE 1 How many ways can you select 3 committee members from a group of 5 people? Is order important? n! C = n r (n − r )! r !
15. 15. EXAMPLE 1 How many ways can you select 3 committee members from a group of 5 people? Is order important? n! C = n r 5 C3 (n − r )! r !
16. 16. EXAMPLE 1 How many ways can you select 3 committee members from a group of 5 people? Is order important? n! 5! C = n r C = 5 3 (n − r )! r ! (5 − 3)!3!
17. 17. EXAMPLE 1 How many ways can you select 3 committee members from a group of 5 people? Is order important? n! 5! 5! C = n r C = 5 3 = (n − r )! r ! (5 − 3)!3! 2 !3!
18. 18. EXAMPLE 1 How many ways can you select 3 committee members from a group of 5 people? Is order important? n! 5! 5! (5)(4)(3)(2)(1) C = n r C = 5 3 = = (n − r )! r ! (5 − 3)!3! 2 !3! (2)(1)(3)(2)(1)
19. 19. EXAMPLE 1 How many ways can you select 3 committee members from a group of 5 people? Is order important? n! 5! 5! (5)(4)(3)(2)(1) C = n r C = 5 3 = = (n − r )! r ! (5 − 3)!3! 2 !3! (2)(1)(3)(2)(1) (5)(4) = 2
20. 20. EXAMPLE 1 How many ways can you select 3 committee members from a group of 5 people? Is order important? n! 5! 5! (5)(4)(3)(2)(1) C = n r C = 5 3 = = (n − r )! r ! (5 − 3)!3! 2 !3! (2)(1)(3)(2)(1) (5)(4) 20 = = 2 2
21. 21. EXAMPLE 1 How many ways can you select 3 committee members from a group of 5 people? Is order important? n! 5! 5! (5)(4)(3)(2)(1) C = n r C = 5 3 = = (n − r )! r ! (5 − 3)!3! 2 !3! (2)(1)(3)(2)(1) (5)(4) 20 = = =10 2 2
22. 22. EXAMPLE 1 How many ways can you select 3 committee members from a group of 5 people? Is order important? n! 5! 5! (5)(4)(3)(2)(1) C = n r C = 5 3 = = (n − r )! r ! (5 − 3)!3! 2 !3! (2)(1)(3)(2)(1) (5)(4) 20 = = =10 ways 2 2
23. 23. EXAMPLE 2 How many ways are there to select 5 people for a committee that has 5 openings?
24. 24. EXAMPLE 2 How many ways are there to select 5 people for a committee that has 5 openings? 5 C5
25. 25. EXAMPLE 2 How many ways are there to select 5 people for a committee that has 5 openings? 5! C = 5 5 (5 − 5)! 5!
26. 26. EXAMPLE 2 How many ways are there to select 5 people for a committee that has 5 openings? 5! 5! C = 5 5 = (5 − 5)! 5! 0 ! 5!
27. 27. EXAMPLE 2 How many ways are there to select 5 people for a committee that has 5 openings? 5! 5! 5! C = 5 5 = = (5 − 5)! 5! 0 ! 5! 5!
28. 28. EXAMPLE 2 How many ways are there to select 5 people for a committee that has 5 openings? 5! 5! 5! C = 5 5 = = =1 way (5 − 5)! 5! 0 ! 5! 5!
29. 29. COMBINATIONS CHECK Are the following possible? If not, why not? a. 5C6 b. 5C−2 c. 10.5C6
30. 30. COMBINATIONS CHECK Are the following possible? If not, why not? a. 5C6 No, can’t choose more than what is available b. 5C−2 c. 10.5C6
31. 31. COMBINATIONS CHECK Are the following possible? If not, why not? a. 5C6 No, can’t choose more than what is available b. 5C−2 No, can’t choose a negative number of things c. 10.5C6
32. 32. COMBINATIONS CHECK Are the following possible? If not, why not? a. 5C6 No, can’t choose more than what is available b. 5C−2 No, can’t choose a negative number of things c. 10.5C6 No, can’t have half of an item
33. 33. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces?
34. 34. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces P(2 aces) = total combinations
35. 35. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces C 4 2 P(2 aces) = = total combinations C 52 2
36. 36. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces C 4 2 P(2 aces) = = total combinations C 52 2 4! C = 4 2 (4 − 2)! 2 !
37. 37. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces C 4 2 P(2 aces) = = total combinations C 52 2 4! 4! C = 4 2 = (4 − 2)! 2 ! 2 ! 2 !
38. 38. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces C 4 2 P(2 aces) = = total combinations C 52 2 4! 4! (4)(3) C = 4 2 = = (4 − 2)! 2 ! 2 ! 2 ! 2
39. 39. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces C 4 2 P(2 aces) = = total combinations C 52 2 4! 4! (4)(3) C = 4 2 = = =6 (4 − 2)! 2 ! 2 ! 2 ! 2
40. 40. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces C 4 2 P(2 aces) = = total combinations C 52 2 4! 4! (4)(3) C = 4 2 = = =6 (4 − 2)! 2 ! 2 ! 2 ! 2 52 ! C = 4 2 (52 − 2)! 2 !
41. 41. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces C 4 2 P(2 aces) = = total combinations C 52 2 4! 4! (4)(3) C = 4 2 = = =6 (4 − 2)! 2 ! 2 ! 2 ! 2 52 ! 52 ! C = 4 2 = (52 − 2)! 2 ! 50 ! 2 !
42. 42. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces C 4 2 P(2 aces) = = total combinations C 52 2 4! 4! (4)(3) C = 4 2 = = =6 (4 − 2)! 2 ! 2 ! 2 ! 2 52 ! 52 ! (52)(51) C = 4 2 = = (52 − 2)! 2 ! 50 ! 2 ! 2
43. 43. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces C 4 2 P(2 aces) = = total combinations C 52 2 4! 4! (4)(3) C = 4 2 = = =6 (4 − 2)! 2 ! 2 ! 2 ! 2 52 ! 52 ! (52)(51) C = 4 2 = = =1326 (52 − 2)! 2 ! 50 ! 2 ! 2
44. 44. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces C 4 2 6 P(2 aces) = = = total combinations 52 C2 1326 4! 4! (4)(3) C = 4 2 = = =6 (4 − 2)! 2 ! 2 ! 2 ! 2 52 ! 52 ! (52)(51) C = 4 2 = = =1326 (52 − 2)! 2 ! 50 ! 2 ! 2
45. 45. EXAMPLE 3 A card is drawn at random from a standard deck of playing cards and is set aside. A second card is drawn. What is the probability of drawing two aces? ways to draw 2 aces C 4 2 6 1 P(2 aces) = = = = total combinations 52 C2 1326 221 4! 4! (4)(3) C = 4 2 = = =6 (4 − 2)! 2 ! 2 ! 2 ! 2 52 ! 52 ! (52)(51) C = 4 2 = = =1326 (52 − 2)! 2 ! 50 ! 2 ! 2
46. 46. HOMEWORK
47. 47. HOMEWORK p. 180 #1-25 odd “You cannot run away from a weakness; you must sometimes ﬁght it out or perish. And if that be so, why not now, and where you stand?” - Robert Louis Stevenson