Counting Principles

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Counting Principles
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Counting examples

• Ex 1 A small sandwich café has 4 different types
  of bread, 5 different types of meat and 3
  different types of cheese. If a sandwich is made
  by one bread, one meat and one cheese, how many
  different sandwiches can the café make?
• Solution (4)(5)(3)60
• Ex 2 A businessman wants to travel from Houston
  to Dallas by either airplane or bus. There are 5
  different airlines and 4 different bus companies
  that offer tickets. How many different travel
  plans are there for the businessman?
• Solution 549

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Basic Counting Principles

• Multiplication Principle
• Event A can occur in m different ways, and after
  event A has occurred, event B can occur in n
  different ways. Then the number of ways that the
  two events occur is mn.
• Addition Principle
• Let event A and B be mutually exclusive events.
  If event A can occur in m different ways and
  event B can occur in n different ways, then the
  number of ways that event A or B occurs is mn.
• Two events are mutually exclusive if they have no
  outcomes in common. (aka they cannot both
  happen)

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Permutation

• A permutation is an arrangement of n unique
  elements in a definite order.
• Ex permutations of letters a, b, c
• abc, acb, bac, bca, cab, cba
• Total of 6 different ways of arrangements. (aka
  6 permutations)

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Factorial

• n! n(n-1)(n-2)(3)(2)(1)
• Ex How many ways can you arrange the three
  distinct letters a, b and c?
• Solution 3! 6

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Example Permutation

• A classroom has 20 students. A three people
  committee contains a president, a vice president
  and a spirit icon. How many different ways can
  this committee be formed?
• Solution (20)(19)(18) 6840

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Permutation

• It is useful, on occasion, to order a subset of a
  collection of elements rather than the entire
  collection. For example, you might want to choose
  and order r elements out of a collection of n
  elements. Such an ordering is called a
  permutation of n elements taken r at a time.

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Combination

• When you count the number of possible
  permutations of a set of elements, order is
  important. As a final topic in this section, you
  will look at a method of selecting subsets of a
  larger set in which order is not important. Such
  subsets are called combinations of n elements
  taken r at a time.
• For instance, the combinations A,B,C and
  B,C,A are equivalent because both sets contain
  the same three elements, and the order in which
  the elements are listed is not important. So, you
  would count only one of the two sets.

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Permutation vs Combination

• From a group of 10 people, 3 will receive prizes.
  One will receive 30, one will receive 20 and
  one will receive 10. How many different ways
  can the three people be chosen to receive these
  prizes?

• From a group of 10 people, 3 will receive prizes.
  The three people will each receive 20. How
  many different ways can the three people be
  chosen to receive these prizes?

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Combination

• It reads as n choose r.
• Other notations

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Example Combinations

• A standard poker hand consists of five cards
  dealt from a deck of 52. How many different
  poker hands are possible? (After the cards are
  dealt, the player may reorder them, and so order
  is not important.).

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Example Combinations

• You are forming a 12-member swim team from 10
  girls and 15 boys. The team must consist of five
  girls and seven boys. How many different
  12-member teams are possible?

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Combination

• Some very important properties of combination

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Five Special Counting Cases
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Case 1 Adjacent Objects

• How many different ways can the letters A, B, C,
  D and E be arranged if B has to follow right
  behind A?
• Trick Treat AB as one object.
• Solution 4! 24

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Example

• A family of 6 members are to be seated in a row
  of 6 chairs.
• How many ways can they sit if the mom and dad
  want to sit next to each other?
• Solution (2!)(5!) 240
• How many ways can they sit if the mom and dad do
  NOT want to sit next to each other?
• Solution 6! 240 480

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Case 2 Alike Objects

• In how many distinguishable ways can the letters
  in BANANA be written?
• Note There are 3 As and 2 Ns, they are not
  distinct letters.

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Alike Objects
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Example

• How many ways can the letters in MISSISSIPPI be
  arranged?
• Total of 11 letters
• Repeated letters 4 I, 4 S, 2 P.

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Example

• If you start from point A, and you can only move
  either up or right for one unit each time. How
  many different paths are there to point B?

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Case 3 Flippable Strings

• Four small cubes of colours red, green, blue and
  black are glued together to form a colourful
  rectangular prism. How many ways can these four
  cubes be glued together?
• Note Same arrangement after flipping
• Trick (n!)/2
• Solution (4!)/2 12

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Example

• Six pieces of pearls of different shapes are
  connected together by a string to form a little
  hanging string for decoration purpose. In how
  many ways can this string be made?
• Solution (6!)/2 360

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Case 4 Round Table

• Four people are sitting around a round table for
  dinner. In how many ways can these four people
  be seated?
• Note Same arrangement after rotation
• Trick (n-1)!
• Solution (4-1)! 6

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Example

• Eight people are sitting around a table for a
  poker game with their poker faces on. How many
  ways can these eight people be seated in the
  game?
• Solution (8-1)! 5040

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Case 5 Keychain

• How many ways can 4 keys be arranged on a
  keychain ring?
• Note Flippable round table
• Trick (n-1)!/2
• Solution (4-1)!/2 3
• Special note Please do not put only one key on a
  keychain ring.

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Example

• Seven pieces of different pearls are placed
  around a circular wire to form a necklace. In
  how many ways can this necklace be made?
• Solution (7-1)!/2 360