Kutztown University of Pennsylvania
Department of Mathematics
MAT017 – Introduction to Mathematics
Spring Term 2008
Chapter 13.1 Notes on
Counting
In class, it was determined that there were eight (8) possible ways of having three
children in the family.
A list of outcomes is given by the sample space
S
:
S
={
bbb
,
bbg
,
bgb
,
bgg
,
gbb
,
gbg
,
ggb
,
ggg
}, and the cardinal number of set
S
,
n
(
S
) = 8.
By actually writing them down, we were able to
count
them.
Also, how about if both
parents want six children and want to determine all possible outcomes?
We can start by
writing each outcome down as we did before, but this approach will be tedious, but
nevertheless, the sample space can be found.
Rather, we are going to look at a method called “Counting” to determine a systematic
way of determining the number of ways this particular family can have six children.
It is
very fast and efficient.
The Fundamental Counting Principle (FCP)
“If we wish to perform a series of tasks and the first task can be done in
a
ways
, the
second can be done in
b
ways
, the third can be done in
c
ways
, and so on, then all tasks
can be done in
a
x
b
x
c
x …. ways
.”
So, let’s investigate the original threechild problem to determine how the eight was
determined through this method of FCP.
Example 1
In the first child, there are 2 (which equals
a
) ways (that is, sexes possible), for the
second sibling, there are also 2 (which equals
b
) sexes possible, and for the third, there
are also 2 (which equals
c
) sexes possible.
So, according to the method of FCP,
n
(
S
) =
a
x
b
x
c
= 2 x 2 x 2 = 8 ways or outcomes.
Example 2
For the sixchild problem, using similar arguments for the threechild case,
a
= 2,
b
= 2,
c
= 2,
d
= 2,
e
= 2, and
f
= 2.
So, the cardinal number for this set, say
A
,
n
(
A
) = 2 x 2 x 2 x 2 x 2 x 2 = 2
6
= 64.
I guess we can list all possible outcomes, but this will be very tedious.
Anyways, here
are some elements/outcomes to set
A
:
A
= {
bbbbbb
,
bbbbbg
,
bbbbgb
,
bbbgbb
, …,
ggggggg
}.
Instructor:
Dr. Perry Lee
1
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Kutztown University of Pennsylvania
Department of Mathematics
MAT017 – Introduction to Mathematics
Spring Term 2008
So, we see that there will be 64 different outcomes from a sixchild family.
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 Spring '08
 Lee
 Math, Counting, Kutztown University of Pennsylvania Department of Mathematics, Mathematics Spring Term, Dr. Perry Lee

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