Chapter 8 Counting Principles: Further Probability Topics

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Chapter 8Counting Principles Further
Probability Topics

• Section 8.2
• Combinations

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Which Counting Technique?

• If the problem involves more than one category or
  repetition, use the multiplication principle of
  counting and multiply the number of choices for
  each category.
• Within any one category, if the order of
  selection is important, use permutations.
• Within any one category, if the order of
  selection is not important, use combinations.

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A tennis squad has 5 members. The coach needs to
select the first singles player and then a second
singles player. In how many ways can he do this?
Since the order in which the players are chosen
is important, we will use a permutation to solve.
5 P 2 20
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• The same tennis coach needs to select a doubles
  team from his five players. How many different
  doubles team does he have to choose from?
• Let the five players be represented by A, B, C,
  D, E.
• Teams AB AC AD AE
• BA BC BD BE
• CA CB CD CE
• DA DB DC DE
• EA EB EC ED

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• Since the team with AB and BA is the same, as
  are others, we can eliminate all those teams that
  are repeats. In other words, since the order
  does not matter, we only write down a combination
  of the players one time.
• Teams AB AC AD AE
• BA BC BD BE
• CA CB CD CE
• DA DB DC DE
• EA EB EC ED

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• Teams AB AC AD AE
• BC BD BE
• CD CE
• DE
• We now have 10 different doubles teams.
• This is an example of a combination problem.

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Combinations

• A subset of items listed without regard to order
  is called a combination.
• Like permutations, repetitions are not allowed in
  combinations.
• Clue words group, committee, set, sample, team
• Combinations are denoted by the notation nCr or
  

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• In how many ways can you construct a 5-person
  committee out of 30 people?
• You have five places left for eight stamps in
  your stamp book. How many different ways can you
  select the five to place in your stamp book?
• There are 10 chips in a bag numbered from 1 to
  10. Four chips are selected at random. How many
  different ways are there of selecting the four
  chips?

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• Suppose that three computer boards in a
  production run of forty are defective. A sample
  of four is to be selected and checked for
  defects.
• How many samples can be chosen?
• How many samples will contain at least one
  defective board?
• In how many ways can a sample of five chocolates
  be selected from a box of twenty-four chocolates?

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• Suppose you have a group of 10 children
  consisting of 4 girls and 6 boys.
• How many four-person teams can be chosen that
  consist of two girls and two boys?
• How many four-person teams contain at least one
  girl?
• In how many ways can a five-card hand consisting
  of three diamonds be dealt from a standard deck
  of 52 cards?

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• How many ways can a student choose eight
  questions from a twelve-question exam if at least
  three questions must be chosen from the first
  five and three questions from the last seven?
• Two co-captains are to be selected from the
  starting five for a basketball team. In how many
  ways can this be done?
• The student association each year selects a
  council consisting of 7 members. If there are 10
  candidates for the 7-member council, how many
  different councils may be elected?
• How many different poker hands can be dealt from
  a standard deck of 52 cards?

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• How many committees can be selected from four
  teachers and 100 students if each committee must
  have two teachers and three students?
• If the Xerox Corporation has to transfer four of
  its 10 junior executives to a new location, in
  how many ways can the four executives be chosen?
• A newspaper boy discovers while delivering his
  papers that he doesnt have enough papers. He
  has eight houses left to deliver to, but only
  five papers left. In how many ways can he
  deliver the remaining newspapers?

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• Alice has a penny, a nickel, a dime, a quarter,
  and a half-dollar. She may spend any three
  coins.
• In how many ways can Alice do this?
• What is the most money she can spend using just
  three coins?
• Joe has to take a math exam that consists of 10
  questions. He must answer only seven of the 10
  questions.
• In how many ways can Joe choose the seven
  questions?
• If he must answer the first and last questions
  and still answer a total of only seven, in how
  many ways can he do this?