Lesson 18: Counting

1
Lesson 18 Counting

• Objectives
• Use the product rule and sum rule to count
  objects
• Count 1-to-1 functions
• Count the number of possible IPv4 internet
  addresses
• Use the principle of inclusion-exclusion for
  counting
• Use tree diagrams for counting

• Outline
• Recursive algorithms
• Iterative algorithms
• Reading 4.1
• Due Weds 4/5
• 3.4 4,8, 11, 18, 26a, 28a, 34, 35

2
Counting (chapter 4)

• Importance of counting
• Allocation of Resources
• Probability
• Security
• Collisions

3
Counting

• Product Rule
• If there are n ways to make choice 1 and m ways
  to make choice 2, then the total number of
  choices is
• mn

4
Computer passwords

• If lowercase alphabetic characters are allowed,
  how many 8-character passwords are there?
• If upper- and lower-case and numeric characters
  are allowed, how many 10-character passwords are
  there?

5
MAC Addresses

• Ethernet addresses consist of a 48-bit MAC
  address hard-coded into each network card. How
  many possible unique cards can be produced with
  this system?
• If the first 24 bits uniquely identifies the
  vendor of the card (which it does), how many
  cards can be associated with each vendors ID?
• How should 3Com respond to this?

6
License Plates

• Assume each state follows this format for their
  license plates
• AIB 346
• how many license plates could be generated
  throughout the US?

7
Sum Rule

• If a first task can be done n ways and a second
  task m ways, but the tasks cannot be done at the
  same time, there are nm choices.

8
Sum Rule

• A character used in a certain font can either be
  an upper-case Latin character, lower-case English
  character, upper-case Greek character, or a
  Hiragana character.
• How many possible characters are there?

9
IPv4

• How many internet addresses are possible?
• Class A 0, 7-bit netid, 24-bit hostid
• Class B 10, 14-bit netid, 16-bit hostid
• Class C 110, 21-bit netid, 8-bit hostid
• (Class D, E, unassigned)
• Class A Cannot have 1111111 as netid
• all 1s or 0s not allowed for any hostid

10
Inclusion-Exclusion

• When tasks are not mutually exclusive, we need to
  subtract the overcount
• A1 ? A2 A1 A2 - A1 ? A2

11
Inclusion-Exclusion

• How many 8-bit binary strings either start with 1
  or end in 110?

12
Inclusion-Exclusion

• How many numbers 1 to 400 are divisible by 6 or 8?

13
Trees

• How many possible outcomes are there for the NCAA
  tournament?

14
Trees for counting

• Pants come in corduroy, denim, and polyester.
  Sizes available are 30, 32, 34, 36, and 38.
  However, the polyester is available in 40 and 48
  to accommodate big tall men. Colors available
  are brown and black for corduroy, black, blue,
  and white for denim, and chartreuse, turquoise,
  and magenta for polyester. The big tall sizes
  also are available in black.
• How many kinds of pants are there?