1
Combinatorics
–
Combinatorics is the study of the arrangement
of discrete objects
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Who Cares?
•
Counting is important whenever we have finite
resources
•
Examples:
–
Words of memory needed to store …
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Instructions needed to solve …
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Number of hits per second supported by …
–
A password must be between six and eight
characters long.
The characters can be a digit or a
letter (case sensitive).
Each password must include
at least one digit.
How many passwords can we
support?
Introduction
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Summation Notation
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Summation Properties
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Summation Formulas
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(
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(
Useful Counting Cheat
Sheet
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Product Rule
•
If you have two sets, A & B,
A
×
B = A * B
•
Example: How many different bit strings are there of length
seven?
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Answer: 2
7
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Extends (by induction) to sequence of
n
sets
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Sum Rule
•
If you have two disjoint
sets, A & B,
A
∪
B = A + B
•
Example: How many ways are there to elect an ACM
president if the president must be a CS faculty member (from
8) or a CS major (from 250)?
–
Answer: 258
•
Extends (by induction) to
n
mutually exclusive sets
Basic Counting
Principles
–
InclusionExclusion Rule
•
If you have two arbitrary sets, A & B,
A
∪
B = A + B  A
∩
B
•
Example: How many bit strings of length eight start with a 1
or end with the two bits 00?
–
Answer: 2
7
+ 2
6
– 2
5
= 160
•
Subsumes the Sum Rule
•
Extends (by induction) to
n
arbitrary sets (see p. 205 of text)
Consider
Venn
Diagram
Basic Counting
Principles
–
Extended Example: Internet Address (IPv4)
•
Sample: 137.28.109.33 (data.cs.uwec.edu)
•
Translation: 32bits
10001001 00011100 01101101 00100001
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Interpretation
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Restrictions
– 1111111
not available as netid of Class A network
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All
0
s or all
1
s are not allowed as any hostid
•
How many possible computer Internet addresses?
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Note, netid for Class A is 7 bits long, netid for Class B is
14 bits long, and netid for Class C is 21 bits long if you
can’t tell from the above picture.
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 Spring '08
 MichaelR.Wick
 Set Theory, Expert Systems, IP address, Summation, class a, Finite set

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