McCombs Math 381
Basic Counting Principles
1
1.
The Product Rule:
Suppose a procedure can be broken down into a finite
sequence of tasks,
T
1
,
T
2
,
T
3
,
! ! !
,
T
m
, where each task in the sequence can be
performed in a finite number of ways, regardless of how the previous tasks were
done. In particular, suppose
T
1
can be performed in
n
1
ways,
T
2
can be
performed in
n
2
ways,
T
3
can be performed in
n
3
ways, .
..,
T
m
can be performed
in
n
m
ways. Given these criteria,
there are a total of
n
1
!
n
2
!
n
3
! ! ! !
n
m
different
ways to perform the procedure.
2.
The Sum Rule:
Suppose the finite set
T
=
the set containing all possible
ways to perform a given task. Suppose also that
T
=
A
1
!
A
2
!
A
3
!"""!
A
m
,
where
A
1
,
A
2
,
A
3
,
! ! !
,
A
m
is a finite sequence of finite, disjoint sets. Given these criteria,
there are a total of
A
1
+
A
2
+
A
3
+
! ! ! !
+
A
m
different ways to perform task.
3.
A
permutation
is an
ordered
sequence
of elements chosen from a set.
4.
A permutation of length
r
from a set with
n
elements is called an
r
permutation
.
The number of different
r
permutations
from a set with
n
elements
is denoted by
P n
,
r
( )
.
Important Theorem:
Given positive integers
n
and
r
, with
1
!
r
!
n
,
P n
,
r
( )
=
n
r
,
if repetition is allowed
n
!
n
!
r
( )
!
,
if repetition is not allowed
"
#
$
%
$
Note that
n
!
n
!
r
( )
!
=
n n
!
1
( )
n
!
2
( )
"""
n
!
r
+
1
( )
.
5.
A
subset
(or
combination
) is an
unordered
selection
of elements chosen from a set.
6.
A set containing
n
elements will yield a total of
2
n
different subsets.
7.
The number of different
r
combinations
, chosen from a set of
n
elements
is denoted by
C n
,
r
( )
.
Important Theorem:
Given positive integers n and r, with
1
!
r
!
n
,
C n
,
r
( )
=
n
r
!
"
#
$
%
=
n
!
r
!
n
’
r
( )
!
8.
InclusionExclusion:
Given finite sets
A
and
B
,
A
B
A
B
A
B
!
=
+
"
#
.
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View Full DocumentMcCombs Math 381
Basic Counting Principles
2
Examples
1.
How many 8digit numbers are there in base 10 that do not have two consecutive digits
the same? (Note: Leading zeros are not permitted.)
We are constructing a list of length 8 from the set {0,1,2,3,4,5,6,7,8,9}.
Since the leading digit cannot be 0, there are 9 choices for that digit.
2
nd
digit:
9 choices, since we cannot repeat the previous digit.
3
rd
digit:
9 choices, since we cannot repeat the previous digit.
4
th
digit:
9 choices, since we cannot repeat the previous digit.
.
.
.
.
.
.
8
th
digit:
9 choices, since we cannot repeat the previous digit.
So we have
8
9
43,046,721
=
possibilities.
2.
How many 5digit numbers are there in base 3 arithmetic? (Note: Leading zeros
are permitted.)
We are constructing a list of length 5 from the set {0,1,2}.
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 Summer '11
 NA
 Product Rule, Counting, Natural number, Finite set, Basic Counting Principles

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