Counting_ Permutations_ and Combinations

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   Ice cream activity
Counting, Permutations,
  and Combinations
       Unit 4 Part 1
              Warm Up
 How many different ways can you arrange
  the letters in REBEL?
 How many different ways can you arrange
  the letters in BERRIEN?
       Basic (or Fundamental)
         Counting Principle
 States that if one event can occur in m
  ways, and a second event can occur in n
  ways, then the number of ways that both
  events can occur is m times n ways.
 Example: Warm Up Examples (class
  discussion)
       Basic (or Fundamental)
         Counting Principle
 And…if there is a third event that can
  occur in p ways, the number of ways that
  all three events can occur is m times n
  times p ways.
 This principle can be extended for any
  number of events.
Graphic Organizer #1
     More Basic (or Fundamental)
     Counting Principle examples…
   You are asked to create a password for
    your email account. The password must
    start with 1 digit (0-9) and be followed by
    three letters (A-Z). How many different
    possible passwords are there?



number      letter     letter    letter
                 Solution
 The digit at the beginning has 10 different
  possibilities (0-9)
 The second position can be any letter A-
  Z…26 possibilities
 The third position can be any letter A-
  Z….26 possibilities
 The fourth position can be any letter A-
  Z…26 possibilities
 10 * 26 * 26 * 26 = 175,560
              Example 2
 Now, let’s try that same password…except
  none of the letters can be repeated!
 10 possibilities
 26 possibilities
 25 possibilities
 24 possibilities
 10 * 26 * 25 * 24 = 156,000
    Tell the number of possibilities
 1. You can go to any of three beaches in
                               2 choices to make:
                               Choice 1: 3 choices
  Florida and choose scuba diving, 3 choices
                              Choice 2:
                                            of
  parasailing, or surfing at the beach 9 your
                                    3•3=
  choice. How many different vacation
  options do you have?
 2. You have 10 shirts, 8 pairs of pants,
    3 Choices to make:
  and 5 pairs of shoes. How many different
    Choice 1: 10 shirts
     Choice 2: 8 pants
  outfit combinations could there be?
     Choice 3: 5 shoes

      10•8•5 = 400
 How many different ways can3•2•1 = 6
                                   you
  rearrange the word CAT?
 Try HORSE        5•4•3•2•1 = 120




   Try HIPPOPOTAMUS
                       12•11•10•9•8•7•6•5•4•3•2•
                                   1=
                               479,001,600
   Try your name!
   Complete Counting Principle Practice WS
    (1-5)
               Factorials
 Factorials are indicated by the “!” symbol.
 Whenever you see a ! with a number that
  means you multiply the number with every
  number below it.
 Ex. 5! Means 5 * 4 * 3 * 2 * 1 = 120
 7! Means 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
 3! Means 3 * 2 * 1 = 6
 0! Always equals 1
        Operations with Factorials
          Factorial practice WS
   Example: 1! + (6-2)!



   Example: (7-3)! ÷ 2!



   Example: 6! ÷ 3!
                 Practice
   Complete (3-10)
      What would you do here?
   How many ways can you choose 3
    pictures out of a group of 5 to arrange in
    a row on your wall?

        Because we are
     choosing a group of 3
      from a big group of
         5…we are not
     arranging each of the
     objects…This will take
      a special formula…
       Important Questions
 We are about to learn some formulas. To
  use these formulas, we need to answer
  some questions first.
 How many things are you choosing from?
  This answer will be your “n” in the
  formulas.
 How many things can you choose? This
  will be your “r” in the formulas.
              Identify n and r
   Thirty five people try out for basketball,
    and only 15 can be on the team.
                                 n = 35
                                 r = 15


   You can choose three toppings for your
    pizza. The pizza restaurant now offers 20
    different toppings.
                                 n = 20
                                  r=3
          Does order matter?
 What if 25 people are running a race, and we
  want to know how many possibilities there are
  for the top three finishers. Does order matter?
 Of course it does!


 If you get three vegetables on your plate and
  you get to choose from ten, does order matter?
 No…you still are going to get each of those
  three vegetables.
Permutations and Combinations
 Permutations occur when order does
  matter…like the first three finishers of a
  race or setting your locker combination or
  password
 Combinations occur when order does not
  matter…like what three vegetables to
  choose
             Permutations
 Use when the order of the items is
  important, as in a race or arranging things
 The formula for a permutation is:



            nPr = __ n!__
                   (n-r)!
Complete graphic organizer #3
      on Permutations
Practice Using Permutation
        Formula WS
        Permutation Practice
 How many different ways can a
  chairperson and an assistant chairperson
  be selected for a research project if there
  are seven scientists available?
 What is your “n” value?
 What is your “r” value?
 Use the permutation formula to find your
  answer.
            More practice…
 How many different ways can four books
  be arranged on a shelf if they are selected
  from eight books?
 What is your “n” value?
 What is your “r” value?
 Use the formula and find the total number
  of combinations.
             Combinations
 Use when items represents different
  scenarios, such as selecting two types of
  yogurt from a dairy case from a selection
  of nine
 The formula for a combination is:


         nCr = __ n!__
                r!(n-r)!
Graphic Organizer #4
Combination example problems
 You are to select two types of yogurt from
  the dairy case from a selection of 9.
 What is your “n” value?
 What is your “r” value?
 Use the combination formula to find the
  number of combinations possible.
    Another example problem…
 Select three members from your class to
  work specific homework problems on the
  board.
 What is your “n” value?
 What is your “r” value?
 Use the combination formula to find the
  total number of combinations possible.
Complete Combination
    Practice WS

				
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