1. Trees and Factorials

2. Trees and Factorials
A job (or an experiment) that requires the completion of
many steps is called a multi-step job.

3. Trees and Factorials
A job (or an experiment) that requires the completion of
many steps is called a multi-step job. For example, to make a
cheese omelet,

4. Trees and Factorials
A job (or an experiment) that requires the completion of
many steps is called a multi-step job. For example, to make a
cheese omelet, we need to (a simplified version):
3. get the eggs,

5. Trees and Factorials
A job (or an experiment) that requires the completion of
many steps is called a multi-step job. For example, to make a
cheese omelet, we need to (a simplified version):
3. get the eggs,
4. get the cheese,

6. Trees and Factorials
A job (or an experiment) that requires the completion of
many steps is called a multi-step job. For example, to make a
cheese omelet, we need to (a simplified version):
3. get the eggs,
4. get the cheese,
3. cook the omelet.

7. Trees and Factorials
A job (or an experiment) that requires the completion of
many steps is called a multi-step job. For example, to make a
cheese omelet, we need to (a simplified version):
3. get the eggs,
4. get the cheese,
3. cook the omelet.
So for the job of making a cheese omelet may be viewed as
a three-step job.

8. Trees and Factorials
Each step may have different options for how it can be carried
out.

9. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs,

10. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs, the options might be:
• get them from the refrig.

11. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs, the options might be:
• get them from the refrigerator
• get them from the store

12. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs, the options might be:
• get them from the refrigerator
• get them from the store
• get them from Joe, the neighbor

13. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs, the options might be:
• get them from the refrigerator
• get them from the store
• get them from Joe, the neighbor
To get the cheese (assuming we have none):

14. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs, the options might be:
• get them from the refrigerator
• get them from the store
• get them from Joe, the neighbor
To get the cheese (assuming we have none):
• get it from the store

15. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs, the options might be:
• get them from the refrigerator
• get them from the store
• get them from Joe, the neighbor
To get the cheese (assuming we have none):
• get it from the store
• get it from Joe, the neighbor

16. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs, the options might be:
• get them from the refrigerator
• get them from the store
• get them from Joe, the neighbor
To get the cheese (assuming we have none):
• get it from the store
• get it from Joe, the neighbor
To cook it:

17. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs, the options might be:
• get them from the refrigerator
• get them from the store
• get them from Joe, the neighbor
To get the cheese (assuming we have none):
• get it from the store
• get it from Joe, the neighbor
To cook it:
• do it over our stove

18. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs, the options might be:
• get them from the refrigerator
• get them from the store
• get them from Joe, the neighbor
To get the cheese (assuming we have none):
• get it from the store
• get it from Joe, the neighbor
To cook it:
• do it over our stove
• do it over Joe’s stove

19. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs, the options might be:
• get them from the refrigerator
• get them from the store
• get them from Joe, the neighbor
To get the cheese (assuming we have none):
• get it from the store
• get it from Joe, the neighbor
To cook it:
• do it over our stove
• do it over Joe’s stove
The different ways the omelet job may be completed can be
represent by a “tree”.

20. Trees and Factorials
Each step may have different options for how it can be carried
out. For example, to get the eggs, the options might be:
• get them from the refrigerator
• get them from the store
• get them from Joe, the neighbor
To get the cheese (assuming we have none):
• get it from the store
• get it from Joe, the neighbor
To cook it:
• do it over our stove
• do it over Joe’s stove
The different ways the omelet job may be completed can be
represent by a “tree”. The tree represent all possible ways of
completing the three tasks above.

21. Trees and Factorials
1st step
Get eggs

22. Trees and Factorials
refrig.
Joe
store
1st step
Get eggs

23. Trees and Factorials
refrig.
Joe
store
1st step
Get eggs
2nd step
Get cheese

24. Trees and Factorials
Joe
refrig. store
Joe
Joe
store
store Joe
1st step
Get eggs
store
2nd step
Get cheese

25. Trees and Factorials
Joe
refrig. store
Joe
Joe
store
store Joe
1st step
Get eggs
store
2nd step
Get cheese 3rd step
Cook it

26. Trees and Factorials
our
Joe Joe
our
refrig. store Joe
our
Joe
Joe Joe
store our
Joe
store Joe our
1st step
Joe
Get eggs
store our
Joe
2nd step
Get cheese 3rd step
Cook it

27. Trees and Factorials
our FJO
Joe Joe FJJ
FSO
our
refrig. store Joe FSJ
our JJO
Joe
Joe Joe JJJ
store our JSO
Joe JSJ
store Joe our SJO
1st step
Joe SJJ
Get eggs
our SSO
store
Joe SSJ
2nd step
Get cheese 3rd step
Cook it
The different ways to make the omelet may be listed.

28. Trees and Factorials
our FJO
Joe Joe FJJ
FSO
our
refrig. store Joe FSJ
our JJO
Joe
Joe Joe JJJ
store our JSO
Joe JSJ
store Joe our SJO
1st step
Joe SJJ
Get eggs
our SSO
store
Joe SSJ
2nd step
Get cheese 3rd step
Cook it
The different ways to make the omelet may be listed.
There are 3x2x2 = 12 ways.

29. Trees and Factorials
Theorem (Multiplication Principle of Multi-step Jobs)

30. Trees and Factorials
Theorem (Multiplication Principle of Multi-step Jobs)
A job requires the completion of k-steps.

31. Trees and Factorials
Theorem (Multiplication Principle of Multi-step Jobs)
A job requires the completion of k-steps. Suppose there are
N1 options to complete the 1st step

32. Trees and Factorials
Theorem (Multiplication Principle of Multi-step Jobs)
A job requires the completion of k-steps. Suppose there are
N1 options to complete the 1st step
N2 options to complete the 2nd step

33. Trees and Factorials
Theorem (Multiplication Principle of Multi-step Jobs)
A job requires the completion of k-steps. Suppose there are
N1 options to complete the 1st step
N2 options to complete the 2nd step
…
Nk options to complete the k’th step

34. Trees and Factorials
Theorem (Multiplication Principle of Multi-step Jobs)
A job requires the completion of k-steps. Suppose there are
N1 options to complete the 1st step
N2 options to complete the 2nd step
…
Nk options to complete the k’th step
Then there are N1xN2x..xNk different ways of doing the job.

35. Trees and Factorials
Theorem (Multiplication Principle of Multi-step Jobs ):
A job requires the completion of k-steps. Suppose there are:
N1 options to complete the 1st step
N2 options to complete the 2nd step
…
Nk options to complete the k’th step
Then there are N1xN2x..xNk different ways of doing the job.
Example A.
A sandwich shop has 6 different types of bread, 4 different
types of meat, 5 different types of cheese and 8
different types of dressings. A regular sandwich requires one
of each ingredient. How many different regular sandwiches
are possible?

36. Trees and Factorials
Theorem (Multiplication Principle of Multi-step Jobs ):
A job requires the completion of k-steps. Suppose there are:
N1 options to complete the 1st step
N2 options to complete the 2nd step
…
Nk options to complete the k’th step
Then there are N1xN2x..xNk different ways of doing the job.
Example A.
A sandwich shop has 6 different types of bread, 4 different
types of meat, 5 different types of cheese and 8
different types of dressings. A regular sandwich requires one
of each ingredient. How many different regular sandwiches
are possible?
Ans: 6x4x5x8 = 960 different sandwiches are possible.

37. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1

38. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1

39. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2

40. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2
3! = 3x2x1=6

41. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2
3! = 3x2x1=6

42. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2
3! = 3x2x1=6
4! = 4x3x2x1=24 etc…

43. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2
3! = 3x2x1=6
4! = 4x3x2x1=24 etc…
We define 0! = 1

44. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2
3! = 3x2x1=6
4! = 4x3x2x1=24 etc…
We define 0! = 1
Example B. We are to schedule to interview 4 people at 4
different time slots 1 pm, 2 pm, 3 pm and 4 pm. How many
different lineups of the interviews are possible?

45. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2
3! = 3x2x1=6
4! = 4x3x2x1=24 etc…
We define 0! = 1
Example B. We are to schedule to interview 4 people at 4
different time slots 1 pm, 2 pm, 3 pm and 4 pm. How many
different lineups of the interviews are possible?
Ans: There are 4 steps to set a schedule:

46. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2
3! = 3x2x1=6
4! = 4x3x2x1=24 etc…
We define 0! = 1
Example B. We are to schedule to interview 4 people at 4
different time slots 1 pm, 2 pm, 3 pm and 4 pm. How many
different lineups of the interviews are possible?
Ans: There are 4 steps to set a schedule:
chose the 1 pm interview  4 ways

47. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2
3! = 3x2x1=6
4! = 4x3x2x1=24 etc…
We define 0! = 1
Example B. We are to schedule to interview 4 people at 4
different time slots 1 pm, 2 pm, 3 pm and 4 pm. How many
different lineups of the interviews are possible?
Ans: There are 4 steps to set a schedule:
chose the 1 pm interview  4 ways
chose the 2 pm interview  3 ways

48. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2
3! = 3x2x1=6
4! = 4x3x2x1=24 etc…
We define 0! = 1
Example B. We are to schedule to interview 4 people at 4
different time slots 1 pm, 2 pm, 3 pm and 4 pm. How many
different lineups of the interviews are possible?
Ans: There are 4 steps to set a schedule:
chose the 1 pm interview  4 ways
chose the 2 pm interview  3 ways
chose the 3 pm interview  2 ways

49. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2
3! = 3x2x1=6
4! = 4x3x2x1=24 etc…
We define 0! = 1
Example B. We are to schedule to interview 4 people at 4
different time slots 1 pm, 2 pm, 3 pm and 4 pm. How many
different lineups of the interviews are possible?
Ans: There are 4 steps to set a schedule:
chose the 1 pm interview  4 ways
chose the 2 pm interview  3 ways
chose the 3 pm interview  2 ways
chose the 4 pm interview  1 way

50. Trees and Factorials
Factorial
Given n a positive integer, we define n factorial as
n! = nx(n -1)x(n – 2)x..x3x2x1
Hence 1! = 1
2! = 2x1= 2
3! = 3x2x1=6
4! = 4x3x2x1=24 etc…
We define 0! = 1
Example B. We are to schedule to interview 4 people at 4
different time slots 1 pm, 2 pm, 3 pm and 4 pm. How many
different lineups of the interviews are possible?
Ans: There are 4 steps to set a schedule:
chose the 1 pm interview  4 ways
chose the 2 pm interview  3 ways
chose the 3 pm interview  2 ways
chose the 4 pm interview  1 way
So there are 4 x 3 x 2 x 1 = 4! = 24 possible line-ups.

51. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.

52. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?

53. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,

54. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,
1st letter  3 options (from three letters)

55. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,
1st letter  3 options (from three letters)
2nd letter  2 options (two letters are left)

56. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,
1st letter  3 options (from three letters)
2nd letter  2 options (two letters are left)
3rd letter  1 option (one letter is left).

57. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,
1st letter  3 options (from three letters)
2nd letter  2 options (two letters are left)
3rd letter  1 option (one letter is left).
Hence, there are 3! = 3x2x1 = 6 arrangements.

58. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,
1st letter  3 options (from three letters)
2nd letter  2 options (two letters are left)
3rd letter  1 option (one letter is left).
Hence, there are 3! = 3x2x1 = 6 arrangements.
Example D. How many different seating arrangements of 7
people in a row of 7 seats are there?

59. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,
1st letter  3 options (from three letters)
2nd letter  2 options (two letters are left)
3rd letter  1 option (one letter is left).
Hence, there are 3! = 3x2x1 = 6 arrangements.
Example D. How many different seating arrangements of 7
people in a row of 7 seats are there?
There are 7 seat, we are to select a person for each seat:

60. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,
1st letter  3 options (from three letters)
2nd letter  2 options (two letters are left)
3rd letter  1 option (one letter is left).
Hence, there are 3! = 3x2x1 = 6 arrangements.
Example D. How many different seating arrangements of 7
people in a row of 7 seats are there?
There are 7 seat, we are to select a person for each seat:
7 x
7 options

61. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,
1st letter  3 options (from three letters)
2nd letter  2 options (two letters are left)
3rd letter  1 option (one letter is left).
Hence, there are 3! = 3x2x1 = 6 arrangements.
Example D. How many different seating arrangements of 7
people in a row of 7 seats are there?
There are 7 seat, we are to select a person for each seat:
7 x 6 x
7 options 6 options

62. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,
1st letter  3 options (from three letters)
2nd letter  2 options (two letters are left)
3rd letter  1 option (one letter is left).
Hence, there are 3! = 3x2x1 = 6 arrangements.
Example D. How many different seating arrangements of 7
people in a row of 7 seats are there?
There are 7 seat, we are to select a person for each seat:
7 x 6 x 5 x
7 options 6 options 5 options

63. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,
1st letter  3 options (from three letters)
2nd letter  2 options (two letters are left)
3rd letter  1 option (one letter is left).
Hence, there are 3! = 3x2x1 = 6 arrangements.
Example D. How many different seating arrangements of 7
people in a row of 7 seats are there?
There are 7 seat, we are to select a person for each seat:
7 x 6 x 5 x 4 x 3 x 2 x 1
7 options 6 options 5 options

64. Trees and Factorials
The multi-step jobs where the number of options decreases by
one as the next step is carried out have answers related to n!.
Example C. How many different arrangements of the letters in
the word “EAT” are there?
There are three letters, we select one letter one at a time,
1st letter  3 options (from three letters)
2nd letter  2 options (two letters are left)
3rd letter  1 option (one letter is left).
Hence, there are 3! = 3x2x1 = 6 arrangements.
Example D. How many different seating arrangements of 7
people in a row of 7 seats are there?
There are 7 seat, we are to select a person for each seat:
7 x 6 x 5 x 4 x 3 x 2 x 1
7 options 6 options 5 options
So there are 7! = 5040 possibilities.

65. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.

66. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! =
4!

67. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1

68. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1

69. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1
= 9x8x7x6x5

70. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1
= 9x8x7x6x5
= 15120

71. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1
= 9x8x7x6x5
= 15120
b. 12!
4!x8!

72. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1
= 9x8x7x6x5
= 15120
b. 12! = 12x11x…8x7x..x4x3x2x1
4!x8! 4x3x2x1x8x7x..x2x1

73. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1
= 9x8x7x6x5
= 15120
b. 12! = 12x11x…8x7x..x4x3x2x1
4!x8! 4x3x2x1x8x7x..x2x1

74. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1
= 9x8x7x6x5
= 15120
b. 12! = 12x11x…8x7x..x4x3x2x1
4!x8! 4x3x2x1x8x7x..x2x1
= 12x11x10x9
4x3x2x1

75. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1
= 9x8x7x6x5
= 15120
b. 12! = 12x11x…8x7x..x4x3x2x1
4!x8! 4x3x2x1x8x7x..x2x1
= 12x11x10x9
4x3x2x1

76. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1
= 9x8x7x6x5
= 15120
b. 12! = 12x11x…8x7x..x4x3x2x1
4!x8! 4x3x2x1x8x7x..x2x1
5
= 12x11x10x9
4x3x2x1

77. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1
= 9x8x7x6x5
= 15120
b. 12! = 12x11x…8x7x..x4x3x2x1
4!x8! 4x3x2x1x8x7x..x2x1
5
= 12x11x10x9
4x3x2x1
= 11x5x9
= 495

78. Trees and Factorials
When dividing factorials, always cancel as much as possible
first.
Example E. Simplify
a. 9! = 9x8x7x..x4x3x2x1
4! 4x3x2x1
= 9x8x7x6x5
= 15120
b. 12! = 12x11x…8x7x..x4x3x2x1
4!x8! 4x3x2x1x8x7x..x2x1
5
= 12x11x10x9
4x3x2x1
= 11x5x9
= 495

79. Trees and Factorials
Exercise A. Draw a tree to represent all possible outcomes for
each of the following multistep jobs. List all the possible
ordered outcomes using the tree you drew. How many
outcomes are there?
A die–roll has six outcomes 1, 2,.., 6
A coin–flip has two possible outcomes H (heads) or T (tails)
1. Flip a coin twice.
2. Flip a coin three times.
3. Roll a die then flip a coin once.
4. Flip a coin, then roll a die then flip a coin again.
5. A die has the numbers {1, 2} colored Red, {3, 4} colored
Green, and {5, 6} colored Blue. We are to roll the die twice and
observe the ordered–colors of the two rolls.
6. As in problem 5 but we note the color of the 1st roll and
the number for the 2nd roll.

80. Trees and Factorials
7. We are to fly from A to B then travel from B to C. There are
three possible flights from A to B and from B to C, it is only
possible by a helicopter, or by a 4–wheel drive, or a dog sled.
List all the possible ways we get accomplish this. How many
possibilities are there?
Exercise B. How many tcomes are there?
3. Roll a die then flip a coin once.
4. Flip a coin, then roll a die then flip a coin again.
5. A die has the numbers {1, 2} colored Red, {3, 4} colored
Green, and {5, 6} colored Blue. We are to roll the die twice and
observe the ordered–colors of the two rolls.
6. As in problem 5 but we note the color of the 1st roll and
the number for the 2nd roll.