Counting Principles

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Counting Principles

• Mulitplication Counting Principle
• Addition Counting Principle

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Multiplication Counting Principle

• If one event can occur in m ways,
• and another event can occur n ways
• Then the number of ways both can occur
  together is m x n.

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Multiplication Counting Principle

• Example 1
• At a store, skateboards are available in 8
  different deck designs. Each deck design is
  available with 4 different wheel assemblies.
• How many skateboard choices does the store offer?

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Ex1 M.C. Princple

• Use the multiplication counting principle
• of diff. designs x of diff. wheels
• 8 x 4
• 32 skateboard choices

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You Try M.C. Principle

• Ex2 Your class is having an election. There are
  4 candidates for president, 6 for vice president,
  3 for secretary, 7 for treasurer.
• How many ways can a president, vice president,
  secretary, and treasurer be chosen?

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EX2 M.C. Principle

• Answer
• 4 president x 6 vice pres. x 3 sec. x 7
  treasurer
• 4 x 6 x 3 x 7
• 504 different ways

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Ex3 Use the multiplication counting principle to
find the number of choices that are available

• a. Choose sneakers, shoes, or sandals in white,
  black, or gray
• Answer 3 x 3 9 different ways
• b. Choose small, medium, large, or extra large
  pants in dark blue, light blue, or black
• Answer 4 x 3 12 different ways

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Addition Counting Principle

• If the possibilities being counted can be divided
  into groups with no possibilities in common, then
  the number of possibilities is the sum of the
  numbers of the possibilities in each group

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Addition Counting Principle

• Example 1 I.D. Cards
• Suppose that each student is assigned an i.d.
  card which contains a unique 4 character (letter
  and digit (number)) barcode. Each barcode
  contains at most 1 digit.
• How many unique i.d. cards are possible?

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Ex1 A.C. Principle

• 0-digits (ALL letters) There are no digits and
  26 choices for each letter.
• 26 x 26 x 26 x 26
• 456,976
• 1-digit (1 num., 3 letters) There are 10
  choices for digits, 26 for letters
• 10 x 26 x 26 x 26
• 175,760
• The digit can be in any of the 4 positions, so 4
  x 175,760
• 703,040
• The last step is to ADD the two totals
• 456,976 703,040 1,160,016

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Ex2 A.C. Principle

• The combination for your gym locker consists of 4
  symbols (letters and digits). If there are 1 or 0
  letters, how many combinations are possible?

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Ex2 A.C. Principle

• 0 letters There are 10 digit choices
• 10 x 10 x 10 x 10
• 10,000
• 1 letter There are 26 letter choices
• 26 x 10 x 10 x 10
• 26,000
• But remember that the letter can be in any
  position
• So.. 4 x 26,000
• 104,000
• ADD TOTALS 10,000 104,000 114,000

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Ex 3. You Try A.C. Principle

• You are to create a code for your computer
  password. The code must consist of 3 symbols
  (letters and digits). How many password
  combinations are possible if at most one digit is
  used?

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Ex3 you try

• 0 digits
• 26 x 26 x 26
• 17,576
• 1 digit
• 10 x 26 x 26
• 6,760
• DIGIT CAN BE IN ANY 3 POSITIONS
• 6,760 X 3 20,280
• ADD
• 20,280 6,760 27,040