Combinatorics

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Combinatorics

• Math 309
• 8/29/2003
• Sec. 2.4

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Combinatorics

• Math 309

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Combinatorics

• Basic Principle of Counting
• (a.k.a. Multiplication Principle)
• Permutations
• Permutations with indistinguishable objects
• Combinations

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Basic Counting Principle

• If experiment 1 has m outcomes and experiment 2
  has n outcomes, then there are mn outcomes for
  both experiments.
• The principle can be generalized for r
  experiments. The number of outcomes of r
  experiments is the product of the number of
  outcomes of each experiment.

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• We define experiment as a means of making an
  observation (e.g. flip a coin, choose a color).
• Each experiment could be making a choice from a
  different set.

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Permutations

• of arrangements of one set,
• order matters
• application of the basic counting principle where
  we return to the same set for the next selection
• P(n,r) n!/(n-r)!

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Permutations with Indistinguishable Objects

• Order the objects as if they were distinguishable
• Then divide out those arrangements that look
  identical.

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Combinations

• the number of selections, order doesnt matter
• C(n,r) n!/(n-r)!r!
• the number of arrangements can be counted by
  selecting the objects and then ordering them
• i.e. P(n,r) C(n,r)r!

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Observations about Combinations

• C(n, r) C(n, n-r)
• C(n, n) C(n, 0) 1
• C(n, 1) n C(n, n-1)
• C(n, 2) n(n-1)/2

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Combining Counting Techniques

• If we are careful with language,
• when we say AND, we multiply
• AND ? multiplication ? intersection
• when we say OR, we add
• OR ? addition ? union