10.1 – Counting by Systematic Listing
One-Part Tasks
The results for simple, one-part tasks can often be listed easily.
Heads or tails
Tossing a fair coin:
Rolling a single fair die
1, 2, 3, 4, 5, 6
Consider a club N with four members:
N = {Mike, Adam, Ted, Helen}
or
N = {M, A, T, H}
In how many ways can this group select a president?
There are four possible results:
M, A, T, and H.
10.1 – Counting by Systematic Listing
Product Tables for Two-Part Tasks
Determine the number of two-digit numbers that can be
written using the digits from the set {2, 4, 6}.
The task consists of two parts:
1. Choose a first digit
2. Choose a second digit
The results for a two-part task can be pictured in a product
table.
First
Digit
2
4
6
2
22
42
62
Second Digit
4
24
44
64
6
26
46
66
9 possible numbers
10.1 – Counting by Systematic Listing
Product Tables for Two-Part Tasks
What are the possible outcomes of rolling two fair die?
10.1 – Counting by Systematic Listing
Product Tables for Two-Part Tasks
Find the number of ways club N can elect a president and
secretary.
N = {Mike, Adam, Ted, Helen} or N = {M, A, T, H}
The task consists of two parts:
1. Choose a president 2. Choose a secretary
M
Pres.
M
A
T
H
Secretary
A
T
H
MM
MA
MT
MH
AM
AA
AT
AH
TM
TA
TT
TH
HM
HA
HT
HH
12 outcomes
10.1 – Counting by Systematic Listing
Product Tables for Two-Part Tasks
Find the number of ways club N can elect a two member
committee.
N = {Mike, Adam, Ted, Helen} or N = {M, A, T, H}
M
Pres.
M
A
T
H
Secretary
A
T
H
MM
MA
MT
MH
AM
AA
AT
AH
TM
TA
TT
TH
HM
HA
HT
HH
6 committees
10.1 – Counting by Systematic Listing
Tree Diagrams for Multiple-Part Tasks
A task that has more than two parts is not easy to analyze with
a product table. Another helpful device is a tree diagram.
Find the number of three digit numbers that can be written
using the digits from the set {2, 4, 6} assuming repeated digits
are not allowed.
A product table will not work for more than two digits.
Generating a list could be time consuming and disorganized.
10.1 – Counting by Systematic Listing
Tree Diagrams for Multiple-Part Tasks
Find the number of three digit numbers that can be written
using the digits from the set {2, 4, 6} assuming repeated digits
are not allowed.
1st #
2
4
6
2nd #
3rd #
4
6
246
6
4
264
2
6
426
6
2
462
2
4
624
4
2
642
6 possibilities
10.1 – Counting by Systematic Listing
Other Systematic Listing Methods
There are additional systematic ways to produce complete
listings of possible results besides product tables and tree
diagrams.
How many triangles (of any size) are in the figure below?
D
E
One systematic approach is begin with A, and
proceed in alphabetical order to write all 3-letter
combinations (like ABC, ABD, …), then cross
out ones that are not triangles and those that
repeat.
C
F
A
B
Another approach is to “chunk” the figure to
smaller, more manageable figures.
There are 12 triangles.
10.2 – Using the Fundamental Counting Principle
Uniformity Criterion for Multiple-Part Tasks:
A multiple part task is said to satisfy the uniformity criterion if
the number of choices for any particular part is the same no
matter which choices were selected for previous parts.
Uniformity exists:
Find the number of three letter combinations that can be written using the
letters from the set {a, b, c} assuming repeated letters are not allowed.
2 dimes and one six-sided die numbered from 1 to 6 are tossed. Generate
a list of the possible outcomes by drawing a tree diagram.
Uniformity does not exists:
A computer printer allows for optional settings with a panel of five on-off
switches. Set up a tree diagram that will show how many setting are
possible so that no two adjacent switches can be on?
10.2 – Using the Fundamental Counting Principle
Uniformity
Find the number of three letter combinations that can be written using the
letters from the set {a, b, c} assuming repeated letters are not allowed.
1st letter
a
b
c
2nd letter
3rd letter
b
c
abc
c
b
acb
a
c
bac
c
a
bca
a
b
cab
b
a
cba
6 possibilities
10.2 – Using the Fundamental Counting Principle
Uniformity
2 dimes and one six-sided die numbered from 1 to 6 are tossed. Generate
a list of the possible outcomes by drawing a tree diagram.
Die #
1
2
3
4
5
6
Dime
d1
d2
d1
d2
d1
d2
d1
d2
d1
d2
d1
d2
1 d1
1 d2
2 d1
2 d2
3 d1
3 d2
4 d1
4 d2
5 d1
5d2
6 d1
6 d2
12 possibilities
10.2 – Using the Fundamental Counting Principle
Uniformity does not exist
A computer printer is designed for optional settings with a panel of three
on-off switches. Set up a tree diagram that will show how many setting
are possible so that no two adjacent switches can be on? (o = on, f = off)
1st switch
2nd switch
3rd switch
o
o
o
f
o
f
o
f
f
o
f
o
f
f
10.2 – Using the Fundamental Counting Principle
Fundamental Counting Principle
The principle which states that all possible outcomes in a
sample space can be found by multiplying the number of ways
each event can occur.
Example:
At a firehouse fundraiser dinner, one can choose from 2 proteins (beef
and fish), 4 vegetables (beans, broccoli, carrots, and corn), and 2 breads
(rolls and biscuits). How many different protein-vegetable-bread
selections can she make for dinner?
Proteins
2
Vegetables

4
Breads

16 possible selections
2
=
10.2 – Using the Fundamental Counting Principle
Example
At the local sub shop, customers have a choice of the following: 3 breads
(white, wheat, rye), 4 meats (turkey, ham, chicken, bologna), 6
condiments (none, brown mustard, spicy mustard, honey mustard,
ketchup, mayo), and 3 cheeses (none, Swiss, American). How many
different sandwiches are possible?
Breads
3
Meats

4
Condiments Cheeses
6

3 =

216 possible sandwiches
10.2 – Using the Fundamental Counting Principle
Example:
Consider the set of digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
(a) How many two digit numbers can be formed if repetitions are allowed?
1st digit
9
2nd digit

10
=
90
(b) How many two digit numbers can be formed if no repetitions are allowed?
1st digit
9
2nd digit

9
=
81
(c) How many three digit numbers can be formed if no repetitions are allowed?
1st digit
9
2nd digit

9
3rd digit

8
=
648
10.2 – Using the Fundamental Counting Principle
Example:
(a) How many five-digit codes are possible if the first two digits are letters and
the last three digits are numerical?
1st digit
26
2nd digit
3rd digit
4th digit
10  10 
 26

676000 possible five-digit codes
5th digit
10
(a) How many five-digit codes are possible if the first two digits are letters and
the last three digits are numerical and repeats are not permitted?
1st digit
26
2nd digit
3rd digit
4th digit
10  9
 25


468000 possible five-digit codes
5th digit
8
10.2 – Using the Fundamental Counting Principle
Factorials
For any counting number n, the product of all counting numbers from n
down through 1 is called n factorial, and is denoted n!.
For any counting number n, the quantity n factorial is calculated by:
n! = n(n – 1)(n – 2)…(2)(1).
Definition of Zero Factorial:
0! = 1
Examples:
b) (4 – 1)!
a) 4!
4321
24
3!
321
6
c)
=
54
= 20
10.2 – Using the Fundamental Counting Principle
Arrangements of Objects
Factorials are used when finding the total number of ways to
arrange a given number of distinct objects.
The total number of different ways to arrange n distinct
objects is n!.
Example:
How many ways can you line up 6 different books on a shelf?
6

5

4

3
720 possible arrangements

2

1
10.2 – Using the Fundamental Counting Principle
Arrangements of n Objects Containing Look-Alikes
The number of distinguishable arrangements of n objects,
where one or more subsets consist of look-alikes (say n1 are of
one kind, n2 are of another kind, …, and nk are of yet another
kind), is given by
n!
n1 !n2 !
nk !
.
Example:
Determine the number of distinguishable arrangements of the letters of
the word INITIALLY.
9 letters
9!
3!  2!
with 3 I’s
and 2 L’s
30240 possible arrangements