Combinations of a Set

1. SECTION 4-7
Combinations of a Set

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. VOCABULARY
1. Combination:
2. nCr :

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

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

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

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

10. PERMUTATIONS VS.
COMBINATIONS
If order is important: Permutation
If order is not important: Combination
n!
n
Pr =
(n − r )!

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

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

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

24. EXAMPLE 2
How many ways are there to select 5 people for a
committee that has 5 openings?
5
C5

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. 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. 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. 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. COMBINATIONS CHECK
Are the following possible? If not, why not?
a. 5C6
b. 5C−2
c. 10.5C6

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 =
52 2
(52 − 2)! 2 !

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 =
52 2 =
(52 − 2)! 2 ! 50 ! 2 !

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 =
52 2 = =
(52 − 2)! 2 ! 50 ! 2 ! 2

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 =
52 2 = = =1326
(52 − 2)! 2 ! 50 ! 2 ! 2

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 =
52 2 = = =1326
(52 − 2)! 2 ! 50 ! 2 ! 2

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 =
52 2 = = =1326
(52 − 2)! 2 ! 50 ! 2 ! 2

46. PROBLEM SET

47. PROBLEM SET
p. 180 #1-25
“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