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					                                 “Counting” Continued…

So far…we have looked at:
1. Fundamental Counting Rule –
How many outcomes are possible with 5 rolls of a die?




2. Permutations –
How many distinct ways can you arrange the letters of the
   word Mississippi?




Objective: Chapter 2 - Systematic Lists                     1
                                          Today…

    We will continue to work with Permutations
    and take a look at Combinations and throw in
    a quick review of the Fundamental Counting
    Rule.




Objective: Chapter 2 - Systematic Lists            2
              Quick Review of Fundamental Counting Rule

    For a sequence of two events in which the
    first event can occur m ways and the second
    event can occur n ways, the events together
    can occur a total of m times n ways.

    Example: How many different combinations
    of sandwiches are possible with a choice of 3
    types of bread, 4 types of meat and 3 types
    of cheese?

Objective: Chapter 2 - Systematic Lists                   3
                                          You Try

    How many different combinations are
    possible on a lock with 4 tumblers with the
    digits 0-9 each on them?




Objective: Chapter 2 - Systematic Lists             4
                                          Factorial Rule

    A collection of n different items can be arranged in order n!
    different ways (This factorial rule reflects the fact that the
    first item may be selected n different ways, the second item
    may be selected n-1 ways, and so on).

    Example: You have just started your own airline company
    called Air America. You have one plane for a route
    connecting Austin, Boise, and Chicago. How many routes
    are possible?




Objective: Chapter 2 - Systematic Lists                          5
                                          You Try

    How many different ways can a series of 5
    questions be asked?




Objective: Chapter 2 - Systematic Lists             6
                                     Permutations Rule

    The number of permutations (or sequences) of r items
    selected from n available items (without replacement):
                                              n!
                              nPr         =
                                            (n - r)!
    Example: We are going to conduct a survey. We would like
    to visit every state capital to ask our poll questions. That
    isn’t going to be possible. We are going to visit 4 capitals.
    How many different routes are possible (example:
    Sacramento, Salem, Phoenix, Boise is not the same as
    Boise, Phoenix, Salem, Sacramento)



Objective: Chapter 2 - Systematic Lists                             7
                      Permutations with Duplicate Items

    If there are n items with n1 alike, n2 alike,…,nk alike, the
    number of permutations of all n items is:

                                    n!
                          n1! . n2! .. . . . . . . nk!
    Example: The letters DDDDRRRRR, represent a sequence
    of diet and regular cola. How many ways can we arrange
    those letters/colas?




Objective: Chapter 2 - Systematic Lists                            8
                                          You Try

    You have 3 red marbles, 3 green marbles
    and 4 blue marbles. How many different
    ways can the marbles be arranged?




Objective: Chapter 2 - Systematic Lists             9
                                          Combinations

      The number of combinations of r items selected
      from n different items is:

                                     n!
                            nCr = (n - r )! r!


“n” different items

 “r” items to be selected
 different orders of the same items are not counted

Objective: Chapter 2 - Systematic Lists                  10
                                          Example

    In the New York State lottery, a player wins first prize by
    selecting the correct 6-number combination when 6 different
    numbers from 1 through 51 are drawn. If a player selects
    one particular 6-number combination, find the probability of
    winning (The player need not select the 6 numbers in the
    same order as they are drawn, so order is irrelevant).




Objective: Chapter 2 - Systematic Lists                       11
                                          You Try

    A standard deck of 52 playing cards has 4
    suits with 13 different cards in each suit.
    If the order in which the cards are dealt is not
    important, how many different 5-card hands
    are possible?




Objective: Chapter 2 - Systematic Lists             12
                                          Big Red Flag!!

    When different orderings of the same items
    are counted separately, we have a
    permutation problem, but when different
    orderings of the same items are not counted
    separately, we have a combination problem.




Objective: Chapter 2 - Systematic Lists                    13
                                 Telling the Difference

    Example: How many different ways can you
    arrange the first four hitters on a baseball
    team of 9?



    Now, how many ways can you arrange the
    first four hitters if order matters?



Objective: Chapter 2 - Systematic Lists                   14
                                          Let’s Try Some

Tell whether you would use a combination or a
  permutation:
1. The winner and first, second, and third
  runner up in a contest with 10 finalists.
2. Selecting two of eight employees to attend
  a business seminar.
3. An arrangement of the letters in the word
  algebra?


Objective: Chapter 2 - Systematic Lists                    15
                                          Homework

Counting Worksheet due Thursday.




Objective: Chapter 2 - Systematic Lists              16

				
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