Permutations and Combinations

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Permutations and Combinations
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Tree Diagrams
S1 H1 M1 H2 S2 H1 H2 S1 H1 M2
H2 S2 H1 H2 S1 H1 M3 H2 S2 H1
H2

• A student has the choice of 3 math courses, 2
  science courses, and 2 humanities courses. She
  can only select one course from each area. How
  many course schedules are possible?

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• There are 3x2x2 possible choices.
• The choice of selecting a mathematics course does
  not affect the choice of ways to select a science
  or humanities course. Thus, these three choices
  are called independent events.
• Events that do affect each other are called
  dependent events. (An example of dependent
  events would be the order in which runners finish
  a race. The first place runner affects the
  possibilities for second)

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Basic Counting Principle

• Suppose one event can be chosen in p different
  ways, and another independent event can be chosen
  in q different ways. Then the two events can be
  chosen successively in p times q ways. The
  principle can be extended to any number in
  independent events.

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Permutations

• The arrangement of objects in a certain order in
  called a permutation. In a permutation, the
  order of the objects in very important.
• The symbol P(n,n) denotes the number of
  permutations of n objects taken all at once.
• The symbol P(n,r) denotes the number of
  permutation of n objects taken r at a time.

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Permutations

• P(n,n) n!
• How many different ways can 7 students stand in a
  straight line for a picture. (All students will
  be lined up at the same time)
• P(7,7) 7! 7x6x5x4x3x2x1 5040 ways

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Permutations
In how many ways can you elect a chairperson,
vice-chairperson, treasurer, and secretary,
assuming that one person cannot hold more than
one office from 10 people. This is a permutation
of 10 people being chosen 4 at a time.
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Combinations

• In a combination the order of the items does not
  matter.
• For example if you had to choose 3 books out of 5
  possible the combination of book A, B, and C is
  the same set of books even if you put them in a
  different order.
• You calculate the number of combinations by
  starting out with P(5,3) and divide the result by
  3! because there are 3! ways to arrange three
  books. The answer would be 10 ways.

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Example

• If a gallery wanted to showcase 4 paintings out
  of the 20 they had. How many groups of 4
  paintings can be chosen?
• Does order matter?

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Example

• There are 15 names on the ballot for junior class
  officers. Five will be selected to form a class
  committee.
• How many different committees of 5 can be formed
• In how many ways can a committee of 5 be formed
  if each student has a different responsibility?
• If there are 8 girls and 7 boys on the ballot,
  how many committees of 2 boys and 3girls can be
  formed.