Documents
Resources
Learning Center
Upload
Plans & pricing Sign in
Sign Out

permutations_and_combinations

VIEWS: 126 PAGES: 21

									         Activating Activity
• If ice cream sundaes come in 5 flavors
  with 4 possible toppings, how many
  different sundaes can be made with one
  flavor of ice cream and one topping?




                                           1
       Counting Principle
         Permutations
         Combinations

  How do you use the counting
   principle, permutations and
combinations to predict outcomes
       of different events or
           occurrences?
                               2
The Fundamental Counting
Principle:

If there are a ways for one
activity to occur, and b ways for a
second activity to occur, then
there are a • b ways for both to
occur.


                                      3
                 Examples:
   1. Activities: roll a die and flip a coin
  There are 6 ways to roll a die and 2 ways to
                  flip a coin.
   There are 6 • 2 = 12 ways to roll a die and
                  flip a coin.


2. Activities: draw two cards from a standard
 deck of 52 cards without replacing the cards
    There are 52 ways to draw the first card.
     There are 51 ways to draw the second
                     card.
   There are 52 • 51 = 2,652 ways to draw the
                  two cards.                     4
The Counting Principle also works for more than
                  two activities.
     3. Activities: a coin is tossed five times
          There are 2 ways to flip each coin.
    There are 2 • 2 • 2 • 2 •2 = 32 arrangements of
                  heads and tails.


     4. Activities: a die is rolled four times
          There are 6 ways to roll each die.
  There are 6 • 6 • 6 • 6 = 1,296 possible outcomes.

       Remember: The Counting Principle is easy! Simply
      MULTIPLY the number of ways each activity can occur.   5
A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3
choices of toppings (no butter, butter, extra butter). How many possible
ways can a bag of popcorn be purchased?




Information about girls' ice skates:
  Colors: white, beige, pink, yellow, blue
  Sizes: 4, 5, 6, 7, 8
  Extras: tassels, striped laces, bells
Assuming that all skates are sold with ONE extra, how many
possible arrangements exist?




                                                                             6
Your state issues license plates consisting of letters and
 numbers. There are 26 letters and the letters may be
  repeated. There are 10 digits and the digits may be
 repeated. How many possible license plates can be
   issued with two letters followed by three numbers?




The ice cream shop offers 31 flavors. You order a double-scoop cone. In
how many different ways can the clerk put the ice cream on the cone if you
wanted two different flavors?




                                                                             7
The local Family Restaurant has a daily breakfast special in
which the customer may choose one item from each of the
following groups:
                Breakfast   Accompan
                                            Juice
                Sandwich     iments
                 egg and     breakfast
                                            orange
                   ham        potatoes
                                           cranberry
                 egg and    apple slices
                                            tomato
                  bacon      fresh fruit
                                             apple
                 egg and        cup
                                             grape
                 cheese        pastry

     a.) How many different breakfast specials
     are possible?
     b.) How many different breakfast specials
     without meat are possible?



                                                          8
Permutation:
A set of objects in which position (or order) is important.
To a permutation, the trio of Brittany, Alan and Greg is
DIFFERENT from Greg, Brittany and Alan. Permutations are
persnickety (picky).


Combination:
 A set of objects in which position (or order) is NOT
important.
To a combination, the trio of Brittany, Alan and Greg is THE
SAME AS Greg, Brittany and Alan.




                                                               9
Let's look at which is which:




                 Permutation        versus   Combination

   1. Picking a team captain,           1. Picking three team members
   pitcher, and shortstop from a        from a group.
   group.



   2. Picking your favorite two         2. Picking two colors from a
   colors, in order, from a color       color brochure.
   brochure.



   3. Picking first, second and         3. Picking three winners.
   third place winners.


                                                                        10
  Formulas:

   A permutation is the
   choice of r things from
   a set of n things
   without replacement
   and where the order
   matters.




Remember: n is the total number of objects
  r is the number of objects chosen (want)

                                             11
1.   Compute:   5   P5




2.   Compute:   6P2




3. Find the number of ways to arrange 5 objects that are chosen from a set of 7 different
objects.




4. What is the total number of possible 5-letter arrangements of the letters w, h, i, t, e, if each
letter is used only once in each arrangement?




5. How many different 3-digit numerals can be made from the digits 4, 5, 6, 7, 8 if a digit can
appear just once in a numeral?




                                                                                                      12
1.   Compute:     5   P5     5 · 4 · 3 · 2 · 1 = 120



2.   Compute:     6P2         6 · 5 = 30                   or
                      multiply by two factors
                       of the factorial, starting with 6


3. Find the number of ways to arrange 5 objects that are chosen from a set of 7 different
objects.

     7P5    = 7·6·5·4·3 = 2520          or



4. What is the total number of possible 5-letter arrangements of the letters w, h, i, t, e, if each
letter is used only once in each arrangement?

      5 P5    = 5·4·3·2·1 = 120            or                             or   simply 5!



5. How many different 3-digit numerals can be made from the digits 4, 5, 6, 7, 8 if a digit can
appear just once in a numeral?

     5 P3    = 5·4·3 = 60             or


                                                                                                      13
A permutation is an arrangement of
objects in specific order.
The order of the arrangement is
important!!



                         Consider, four students walking toward their
                       school entrance. How many different ways could
                         they arrange themselves in this side-by-side
                                           pattern?
                                 1,2,3,4   2,1,3,4   3,2,1,4   4,2,3,1
                                 1,2,4,3   2,1,4,3   3,2,4,1   4,2,1,3
                                 1,3,2,4   2,3,1,4   3,1,2,4   4,3,2,1
                                 1,3,4,2   2,3,4,1   3,1,4,2   4,3,1,2
                                 1,4,2,3   2,4,1,3   3,4,2,1   4,1,2,3
                                 1,4,3,2   2,4,3,1   3,4,1,2   4,1,3,2



                  The number of different arrangements is 24 or 4! = 4 •
                   3 • 2 • 1. There are 24 different arrangements, or
                 permutations, of the four students walking side-by-side.
                                                                       14
In how many ways can 3 different vases be arranged on a tray?




                                                                15
A combination is the choice of r things from a set of n
things without replacement and where order does not
matter.




                                                          16
Example 1:
Evaluate     :




                  Notice how the cancellation occurs, leaving only
                 2 of the factorial terms
                  in the numerator. A pattern is emerging ... when
                 finding a combination
                  such as the one seen in this problem, the
                 second value (2) will tell you
                  how many of the factorial terms to use in the
                 numerator, and the
                  denominator will simply be the factorial of the
                 second value (2).




                                                                     17
Heather has finally narrowed her clothing choices for the big
party down to 3 skirts, 2 tops and 4 pair of shoes. How many
     different outfits could she form from these choices?




                                                                18
A coach must choose five starters from a team of 12
players. How many different ways can the coach choose the
starters?




                                                            19
Joleen is on a shopping spree. She buys six tops, three shorts and 4 pairs of sandals. How many
 different outfits consisting of a top, shorts and sandals can she create from her new purchases?
                                     (6)(3)(4) = 72 possible outfits




             What is the total number of possible 4-letter arrangements of the letters
               m, a, t, h, if each letter is used only once in each arrangement?
                                    or                                     or simply 4!




                                                                                                    20
     3-2-1 Ticket out the door
• Name three different terms used to predict
  outcomes of events.

• Name two formulas used today.

• Name one way to tell which concept you
  will choose to use to predict outcomes of
  events.
                                              21

								
To top