Math 30: Pre-Calculus

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F. How Many Ways can You Organize?

• Math 30 Pre-Calculus
• PC30.12
• Demonstrate understanding of permutations,
  including the fundamental counting principle.
  PC30.13
• PC30.13 Demonstrate understanding of combinations
  of elements, including the application to the
  binomial theorem.

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Key Terms
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1. Permutations

• PC30.12
• Demonstrate understanding of permutations,
  including the fundamental counting principle.

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1. Permutations
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• Counting Methods are used to determine the number
  of members from a set as well as the outcome of
  an event.
• There are methods such as tables, lists or tree
  diagrams that allow you to visually see all the
  outcomes.
• Another methods for determining the number of
  outcomes is called the Fundamental Counting
  Principle.

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• The fundamental counting principle states that if
  one task can be performed in a ways and second
  task can be performed in b ways, then the two
  tasks can be performed in (a)(b) ways.

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Example 1
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• In the last example when we arranged the 5
  students in the middle we end up with
  (5)(4)(3)(2)(1) ways.
• This can be abbreviated as 5! and is read as 5
  factorial.
• Therefore, 5! (5)(4)(3)(2)(1)

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• Seven different objects can arranged 7! Ways
• If there are 7 members of a student council how
  many ways can they select the chair, secretary
  and treasurer?

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Example 2
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• With permutations we said order of different
  objects is important. Well what is some of the
  objects in the set are identical.
• Consider the word WEED and all the possible 4
  letter arrangements.

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• If all the letters were different the number of
  outcomes would be 4!24
• There are however 2 identical letters. If they
  were different we would arrange them 2!2 ways.
• So the number of arrangements of the word WEED is
  

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• For permutations with repeating objects, a set of
  n objects with a of one kind that are
  identical, b of a second kind that are
  identical, and c of a third kind that are
  identical, and so one, can be arranged in

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Example 3
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Example 4
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• To solve some problems you must count the
  different arrangements in all the cases that
  together cover all the possibilities.
• Calculate the number of arrangements for each
  case and then add that values for all cases to
  obtain the total number of arrangements.
• Whenever you encounter a situation with
  constraints or restriction, always address the
  choices for the restricted positions first.

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• For example, you may need to determine the number
  of arrangements of 4 girls and 3 boys in a row of
  7 seats if the end of the rows must be either
  both male or both female.

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Example 5
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Key Ideas p. 526
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Practice

• Ex. 11.1 (p.524) 1,2-8 odds in each, 9-18 evens,
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• 5-8 odds in each, 9, 8-26 evens

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2. Combinations

• PC30.13
• PC30.13 Demonstrate understanding of combinations
  of elements, including the application to the
  binomial theorem.

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2. Combinations
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• A combination is a selection of a group of
  objects taken from a larger group.
• The kinds of objects selected is important but
  NOT the order in which they are selected.
• There are a few ways to find the possible number
  of combinations

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• On is to use reasoning. Use the fundamental
  counting principle and divide by the number of
  ways that the objects can be arranged among
  themselves.
• For example, calculate the number of combinations
  of 3 digits made from 1-5 without repetition.

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• There are 60 ways to arrange 3 items form 5
• However, 3 digits can be arranged 3! Ways among
  themselves and in a combination there are
  considered the same selection.
• So

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• So number of ways of choosing 3 digits from five
  digits is

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• The number of combinations of n items taken r
  at a time is equivilent to teh number of
  combinations of n items taken n-r at a time that
  is nCrnCn-r
• Proof

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• To solve some problems, count the different
  combinations in cases that together cover all the
  possibilities.
• Calculate the number of combinations for each
  case and then add the values for all cases to
  obtain the total number of combinations.

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Example 1
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Example 2
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Example 3
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• When answering questions it is important to know
  if you are dealing with a permutation or a
  combination.
• Remember in Permutations the order of the objects
  is important.
• IN the combinations the type of objects is
  important but NOT the order in which they are
  selected.

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Key Ideas p.533
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Practice

• Ex. 11.2 (p.534) 1-6 odds in each, 7-13, 14-20
  evens
• 4-6 odds in each, 7-13, 14-24 evens

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3. The Binomial Theorem

• PC30.13
• PC30.13 Demonstrate understanding of combinations
  of elements, including the application to the
  binomial theorem.

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3. The Binomial Theorem
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• If you expand a power of a binomial expression,
  such as (xy)4 you get a series of terms
• There are many patterns in the expression of
  (xy)4

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Example 1
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• For coefficients you can use Pascals triangle
  instead of Combinations.

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• Important Properties of the binomial expansion
  (xy)n include
• Write binomial expansions in descending order of
  exponent of the first term in the binomial
• The number of objects, k, selected in the
  combination nCk can be taken to match the number
  of factors of the second variable. That is, it
  is the same as the exponent on the second
  variable.
• The sum of the exponents in any term of the
  expansion is n.

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• The General Term tk1 has the form
• nCk (x)n-k(y)k

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Example 2
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Key Ideas p.541
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Practice

• Ex. 11.3 (p.542) 1-7 odds in each, 8-9, 11-19
  odds
• 2-3, 5-7 odds in each, 9-25 odds