McCombs Math 381
More Counting Examples
1.
In a technician's box there are 400 VLSI chips, 12 of which are faulty. How many
ways are there to pick two chips, so that one is a working chip and the other is
faulty?
There are
12
1
!
"
#
$
%
ways to pick a faulty chip. There are
388
1
!
"
#
$
%
ways to pick
a working chip.
Total ways =
12
1
!
"
#
$
%
388
1
!
"
#
$
%
=
12
’
388
=
4656
.
2.
How many truth tables are possible for compound propositions with the five
variables
p
,
q
,
r
,
s
,
t
?
Note that a truth table with five variables must contain
2
5
=
32
rows.
There are 2 possible outcomes for each row, i.e. T or F.
Thus, there are
2
32
possible five variable truth tables.
3.
How many bit strings of length 10 have more 0s than 1s?
If there are more 0s than 1s in the bit string, then there are five possible cases.
six 0s:
10
6
!
"
#
$
%
possibilities
seven 0s:
10
7
!
"
#
$
%
possibilities
eight 0s:
10
6
!
"
#
$
%
possibilities
nine 0s:
10
6
!
"
#
$
%
possibilities
ten 0s:
10
6
!
"
#
$
%
possibilities
Total =
10
6
!
"
#
$
%
+
10
7
!
"
#
$
%
+
10
8
!
"
#
$
%
+
10
9
!
"
#
$
%
+
10
10
!
"
#
$
%
=
1
2
!
"
#
$
%
2
10
’
10
5
!
"
#
$
%
!
"
#
$
%
4.
How many subsets with an odd number of elements does a set with 10 elements
have?
10
1
!
"
#
$
%
+
10
3
!
"
#
$
%
+
10
5
!
"
#
$
%
+
10
7
!
"
#
$
%
+
10
9
!
"
#
$
%
=
2
9
5.
How many subsets with more than two elements does a set with 100 elements
have?
There are a total of
2
100
total subsets. There are
100
0
!
"
#
$
%
+
100
1
!
"
#
$
%
+
100
2
!
"
#
$
%
subsets with cardinality less than or equal to 2. Thus, there are
2
100
!
100
0
"
#
$
%
’
!
100
1
"
#
$
%
’
!
100
2
"
#
$
%
’
subsets with more than two
elements.
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View Full DocumentMcCombs Math 381
More Counting Examples
6.
Each user has a password 6 characters long where each character is an uppercase
letter, a lowercase letter, or a digit. Each password must contain at least one digit.
How long will it take to check every possible character combination, if each
check takes one unit of time.
There are
26
+
26
+
10
( )
6
total password options.
There are
26
+
26
( )
6
passwords that do not contain a digit.
So there are
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 Summer '11
 NA
 Counting, Doughnut, sj

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