Chapter 5 Section 5

1
Chapter 5Section 5

• Counting
• Techniques

2
Chapter 5 Section 5

• Learning objectives
• Solving counting problems using the
  Multiplication Rule
• Solving counting problems using permutations
• Solving counting problems using combinations
• Solving counting problems involving
  permutations with nondistinct items
• Compute probabilities involving permutations
  and combinations

3
Chapter 5 Section 5

• Learning objectives
• Solving counting problems using the
  Multiplication Rule
• Solving counting problems using permutations
• Solving counting problems using combinations
• Solving counting problems involving
  permutations with nondistinct items
• Compute probabilities involving permutations
  and combinations

4
Chapter 5 Section 5

• The classical method, when all outcomes are
  equally likely, involves counting the number of
  ways something can occur
• This section includes techniques for counting the
  number of results in a series of choices, under
  several different scenarios

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Chapter 5 Section 5

• Example
• If there are 3 different colors of paint (red,
  blue, green) that can be used to paint 2
  different types of toy cars (race car, police
  car), then how many different toys can there be?
• 3 colors 2 cars 3 2 6 different toys
• This can be shown in a table or in a tree diagram

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Chapter 5 Section 5

• A table of the different possibilities
• This is a rectangle with 2 rows and 3 columns 2
  3 6 entries

Red Blue Green
Race Car RedRace Car BlueRace Car Green Race Car
Police Car Red Police Car Blue Police Car Green Police Car
7
Chapter 5 Section 5

• A tree diagram of the different possibilities
• This also shows that there are 6 possibilities

Paint
Car
8
Chapter 5 Section 5

• The Multiplication Counting Rule multiply the
  number of different tasks
• Example 3 colors of paint for 2 types of race
  cars.
• Example 4 colors of paint for 3 types of race
  cars with 6 choices of tires.
• Example How many meals are possible with 4
  appetizers, 5 entrees, and 3 desserts?

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Chapter 5 Section 5

• Example Part A
• A child is coloring a picture of a shirt and
  pants
• There are 5 different colors of markers
• How many ways can this be colored?
• By the multiplication rule
• 5 5 25

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Chapter 5 Section 5

• Example Part B
• A child is coloring a picture of a shirt and
  pants
• There are 5 different colors of markers
• The child wants to use 2 different colors
• How many ways can this be colored?
• By the multiplication rule
• 5 4 20

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Chapter 5 Section 5

• Allowing the same marker to be used twice
• 5 5 25
• Requiring that there be two different markers
• 5 4 20
• There are 5 selections for the first choice for
  both Part A and Part B of this example
• But they differ for the second choice there are
  only 4 selections for Part B

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Chapter 5 Section 5

• Example continued
• Part A, allowing the same marker to be used
  twice, is called counting with repetition and has
  formulas such as
• 5 5 5
• Part B, requiring that there be two different
  markers, is called counting without repetition
  and has formulas such as
• 5 4 3

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Chapter 5 Section 5

• Factorial symbol n!
• n! n (n-1) (n-2) 2 1
• We start off by saying that
• 0! 1 and 1! 1
• For example
• 5! 5 4 3 2 1 120
• Notice how 5! looks like the 5 4 3 from the
  previous example

14
Chapter 5 Section 5

• Learning objectives
• Solving counting problems using the
  Multiplication Rule
• Solving counting problems using permutations
• Solving counting problems using combinations
• Solving counting problems involving
  permutations with nondistinct items
• Compute probabilities involving permutations
  and combinations

15
Chapter 5 Section 5

• The problem of choosing one marker out of 5 and
  then choosing a second marker out of the 4
  remaining is an example of a permutation
• A permutation is an ordered arrangement
• Ex. Write out all the arrangements of the
  letters in USA
• The number of ways is written nPr

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Chapter 5 Section 5

• The number of ways of choosing one marker out of
  5 and then choosing a second marker out of the 4
  remaining is 5P2 20

17
Chapter 5 Section 5

• Stop

18
Chapter 5 Section 5

• Learning objectives
• Solving counting problems using the
  Multiplication Rule
• Solving counting problems using permutations
• Solving counting problems using combinations
• Solving counting problems involving
  permutations with nondistinct items
• Compute probabilities involving permutations
  and combinations

19
Chapter 5 Section 5

• For some problems, the order of choice does not
  matter
• Order matters example
• Choosing one person to be the president of a club
  and another to be the vice-president
• Two different roles
• Order does not matter example
• Choosing two people to go to a meeting
• The same role

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Chapter 5 Section 5

• When order does not matter, this is called a
  combination
• A combination is an
• unordered arrangement, in which
• r different objects are chosen out of
• n different objects with
• repetition not allowed
• The number of ways is written nCr

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Chapter 5 Section 5

• Comparing the description of a permutation with
  the description of a combination
• The only difference is whether order matters

Permutation Combination
Order matters Order does not matter
Choose r objects Choose r objects
Out of n objects Out of n objects
No repetition No repetition
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Chapter 5 Section 5

• Example
• If there are 8 researchers and 3 of them are to
  be chosen to go to a meeting permutations or
  combinations?
• A combination since order does not matter
• There are 56 different ways that this can be done

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Chapter 5 Section 5

• Is a problem a permutation or a combination?
• One way to tell
• Write down one possible solution (i.e. Roger,
  Rick, Randy)
• Switch the order of two of the elements (i.e.
  Rick, Roger, Randy)
• Is this the same result?
• If no this is a permutation order matters
• If yes this is a combination order does not
  matter

24
Chapter 5 Section 5

• Learning objectives
• Solving counting problems using the
  Multiplication Rule
• Solving counting problems using permutations
• Solving counting problems using combinations
• Solving counting problems involving
  permutations with nondistinct items
• Compute probabilities involving permutations
  and combinations

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Chapter 5 Section 5

• Our permutation and combination problems so far
  assume that all n total items are different
• Sometimes we have a permutations but not all of
  the n items are different
• This is a more complicated problem
• How many ways are there?

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Chapter 5 Section 5

• Example
• How many ways to put 3 As, 2 Ns, and 2 Ts to
  try to make a seven letter sequence?
• ____ ____ ____ ____ ____ ____ ____
• Each of the blanks can be filled in with either
  an A or a N or a T
• The three As are the same the two Ns are the
  same the two Ts are the same

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Chapter 5 Section 5

• Example continued
• Where can the As go?
• ____ ____ ____ ____ ____ ____ ____
• There are 7 possible places
• Any 3 of them are possible
• Order does not matter
• So 7C3 different ways to put in the As
• What about the Ns and Ts?
• 4C2 different ways to put in the Ns
• 2C2 different ways to put in the Ts

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Chapter 5 Section 5

• Example continued
• Altogether there are
• 7C3 4C2 2C2
• different ways
• This is
• Notice that the denominator is 3, 2, 2 the
  numbers of each letter

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Chapter 5 Section 5

• The general formula for the number of
  permutations of
• n total objects where there are
• n1 of the first kind
• n2 objects of the second kind
• and
• nk of the kth kind is

30
Chapter 5 Section 5

• Learning objectives
• Solving counting problems using the
  Multiplication Rule
• Solving counting problems using permutations
• Solving counting problems using combinations
• Solving counting problems involving
  permutations with nondistinct items
• Compute probabilities involving permutations
  and combinations

31
Chapter 5 Section 5

• A permutation example
• In a horse racing Trifecta, a gambler must pick
  which horse comes in first, which second, and
  which third
• If there are 8 horses in the race, and every
  order of finish is equally likely, what is the
  chance that any ticket is a winning ticket?
• Order matters, so this is a permutations problem
• There are 8P3 permutations of the order of finish
  of the horses
• The probability that any one ticket is a winning
  ticket is 1 out of 8P3, or 1 out of 336

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Chapter 5 Section 5

• A combination example
• The Powerball lottery consists of choosing 5
  numbers out of 55 and then 1 number out of 42
• The grand prize is given out when all 6 numbers
  are correct
• What is the chance of getting the grand prize?
• Order doesnt matter (combinations problem)
• There are 55C5 combinations of the 5 numbers
• There are 42C1 combinations of the last number
• 1/( 55C5 x 42C1 )

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Summary Chapter 5 Section 5

• The Multiplication Rule counts the number of
  possible sequences of items
• Permutations and combinations count the number of
  ways of arranging items, with permutations when
  the order matters and combinations when the order
  does not matter
• Permutations and combinations are used to compute
  probabilities in the classical method