SECTION 4-7
Combinations of a Set
ESSENTIAL QUESTION


• How   do you find the number of combinations of a set?



• Where   you’ll see this:

 • Cooking, tr...
VOCABULARY

1. Combination:


2. nCr :
VOCABULARY

1. Combination: The number of ways you can pick from a
    set of items when order is not important.

2. nCr :
VOCABULARY

1. Combination: The number of ways you can pick from a
    set of items when order is not important.

2. nCr :...
VOCABULARY

1. Combination: The number of ways you can pick from a
    set of items when order is not important.

2. nCr :...
PERMUTATIONS VS.
   COMBINATIONS
  If order is important:
If order is not important:
PERMUTATIONS VS.
   COMBINATIONS
  If order is important: Permutation
If order is not important:
PERMUTATIONS VS.
   COMBINATIONS
  If order is important: Permutation
If order is not important: Combination
PERMUTATIONS VS.
   COMBINATIONS
  If order is important: Permutation
If order is not important: Combination



          ...
PERMUTATIONS VS.
   COMBINATIONS
  If order is important: Permutation
If order is not important: Combination



          ...
EXAMPLE 1
How many ways can you select 3 committee members
           from a group of 5 people?
EXAMPLE 1
How many ways can you select 3 committee members
           from a group of 5 people?

               Is order i...
EXAMPLE 1
How many ways can you select 3 committee members
           from a group of 5 people?

                     Is o...
EXAMPLE 1
How many ways can you select 3 committee members
           from a group of 5 people?

                     Is o...
EXAMPLE 1
How many ways can you select 3 committee members
           from a group of 5 people?

                     Is o...
EXAMPLE 1
How many ways can you select 3 committee members
           from a group of 5 people?

                     Is o...
EXAMPLE 1
How many ways can you select 3 committee members
           from a group of 5 people?

                     Is o...
EXAMPLE 1
How many ways can you select 3 committee members
           from a group of 5 people?

                       Is...
EXAMPLE 1
How many ways can you select 3 committee members
           from a group of 5 people?

                       Is...
EXAMPLE 1
How many ways can you select 3 committee members
           from a group of 5 people?

                      Is ...
EXAMPLE 1
How many ways can you select 3 committee members
           from a group of 5 people?

                      Is ...
EXAMPLE 2
How many ways are there to select 5 people for a
       committee that has 5 openings?
EXAMPLE 2
How many ways are there to select 5 people for a
       committee that has 5 openings?




        5
           ...
EXAMPLE 2
How many ways are there to select 5 people for a
       committee that has 5 openings?



                 5!
  ...
EXAMPLE 2
How many ways are there to select 5 people for a
       committee that has 5 openings?



                 5!   ...
EXAMPLE 2
How many ways are there to select 5 people for a
       committee that has 5 openings?



                 5!   ...
EXAMPLE 2
How many ways are there to select 5 people for a
       committee that has 5 openings?



                 5!   ...
COMBINATIONS CHECK
Are the following possible? If not, why not?

                   a. 5C6


                   b. 5C−2


...
COMBINATIONS CHECK
Are the following possible? If not, why not?

                    a. 5C6
No, can’t choose more than wha...
COMBINATIONS CHECK
 Are the following possible? If not, why not?

                    a. 5C6
No, can’t choose more than wh...
COMBINATIONS CHECK
 Are the following possible? If not, why not?

                    a. 5C6
No, can’t choose more than wh...
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 i...
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 i...
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 i...
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 i...
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 i...
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 i...
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 i...
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 i...
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 i...
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 i...
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 i...
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 i...
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 i...
HOMEWORK
HOMEWORK


                     p. 180 #1-25 odd




“You cannot run away from a weakness; you must sometimes
  fight it ou...
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Integrated Math 2 Section 4-7

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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

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