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Permutations and the Fundamental Counting Principle: Finding Different Arrangements

This task involves using popsicle sticks to explore the concept of permutations and the Fundamental Counting Principle. Students will be asked to determine the number of different outcomes in various situations, such as choosing lunch items or creating greeting cards.

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Permutations and the Fundamental Counting Principle: Finding Different Arrangements

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  1. Task Pick up 1 bag of popsicle sticks per group (4 students per group) #1 With the popsicle sticks, make a list of all the different ways you can choose 2 out of the 3 colors (red, green & yellow) ORDER MATTERS Example #2 With the popsicle sticks, make a list of all the different ways you can choose 2 out of the 4 colors (red, green, yellow, & blue) ORDER MATTERS

  2. Find Probabilities Using Permutations Lesson 6.2 Page 342 (Day 1)

  3. Review: Remember, you can use a tree diagram or The Fundamental Counting Principle to help you find the number of outcomes in a situation.

  4. Use an appropriate method to find the number of outcomes in each of the following situations: chicken, chips, apple chicken, chips, orange chicken, chips, milk apple juice orange juice milk chips fruit apple juice orange juice milk chicken, fruit, apple chicken, fruit, orange chicken, fruit, milk tuna, chips, apple tuna, chips, orange tuna, chips, milk apple juice orange juice milk chips fruit apple juice orange juice milk tuna, fruit, apple tuna, fruit, orange tuna, fruit, milk 1. Your school cafeteria offers chicken or tuna sandwiches; chips or fruit; and milk, apple juice, or orange juice. If you purchase one sandwich, one side item and one drink, how many different lunches can you choose? There are 12 possible lunches. Sandwich(2) Side Item(2) Drink(3) Outcomes chicken tuna

  5. 2. A greeting card software program offers 24 different greetings, 10 different graphic images, and 8 font styles. How many different greeting cards can you make? There are 1,920 possible greeting cards. Greeting(24)Image(10)Font (8)Outcomes In this situation it makes more sense to use the Fundamental Counting Principle. • 24 • 10 8 = 1,920

  6. Drama Comedy Comedy Drama Musical, Drama, Comedy Musical, Comedy, Drama Musical Comedy Comedy Musical Drama, Musical, Comedy Drama, Comedy, Musical Musical Drama Drama Musical Comedy, Musical, Drama Comedy, Drama, Musical 3. Suppose you rent a musical, a drama, and a comedy from your local video store. In how many different orders can you watch the videos? There are 6 possible orders. What if you add an action movie? How many choices would you have for the 1st movie? 2nd? etc.? First MovieSecond MovieThird Movie Outcomes Musical Drama Comedy

  7. Find Probabilities Using Permutations Lesson 6.2 Page 342 (Day 2)

  8. 7 1 • • 6 • 5 • 4 • 3 • 2 • 4. Many mp3 players can vary the order in which songs are played. Your mp3 currently only contains 8 songs. Find the number of orders in which the songs can be played. There are 40,320 possible song orders. 1st Song 2nd3rd4th5th6th7th8thOutcomes In this situation it makes more sense to use the Fundamental Counting Principle. = 8 40,320 The solutions in examples 3 and 4 involve the product of all the integers from n to one. The product of all positive integers less than or equal to a number is a factorial.

  9. Factorial The product of counting numbers beginning at n and counting backward to 1 is written n! and it’s called n factorial. factorial. EXAMPLE 3: (Movies) ‘three factorial’ 3! = 3 • 2 • 1 = 6 EXAMPLE 4: (Songs) ‘eight factorial’ 8! = 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320

  10. Factorial Simplify each expression. 4! 6! c. For the 8th grade field events there are five teams: Red, Orange, Blue, Green, and Yellow. Each team chooses a runner for lanes one through five. Find the number of ways to arrange the runners. 4 • 3 • 2 • 1 = 24 6 • 5 • 4 • 3 • 2 • 1 = 720 5! = 5 • 4 • 3 • 2 • 1 = 120

  11. 5. The student council of 15 members must choose a president, a vice president, a secretary, and a treasurer. There are 32,760 permutations for choosing the class officers. President(15)Vice(14)Secretary (13) Treasurer(12)Outcomes In this situation it makes more sense to use the Fundamental Counting Principle. = 12 • 15 • 14 • 13 32,760

  12. Let’s say the student council members’ names were: John, Miranda, Michael, Kim, Pam, Jane, George, Michelle, Sandra, Lisa, Patrick, Randy, Nicole, Jennifer, and Paul. If Michael, Kim, Jane, and George are elected, would the order in which they are chosen matter? PresidentVice PresidentSecretaryTreasurer Is Michael Kim Jane George the same as… Jane Michael George Kim ? Although the same individual students are listed in each example above, the listings are not the same. Each listing indicates a different student holding each office. Therefore we must conclude that the order in which they are chosen matters.

  13. Permutation When deciding who goes 1st, 2nd, etc., order is important. A permutationis an arrangement or listing of objects in a specific order. The order of the arrangement is very important!!  For example, the number of different ways 15 students can be chosen as class officers can be shown as  15P4 or  15•14•13•12=32,760.  There are 32,760 different arrangements, or permutations, of the fifteen students in which four of them are elected as class officers . The notation for a permutation:   nPr n  is the total number of objects r is the number of objects selected (wanted)

  14. *Note  if  n = r   then =  n! The number of permutations of n objects taken r at a time, where is given by A permutation is an arrangement of objects in which order is important. The number of permutations of n objects is given by

  15. Permutations Simplify each expression. 12P2 b. 10P4 c. At a school science fair, ribbons are given for first, second, third, and fourth place, There are 20 exhibits in the fair. How many different arrangements of four winning exhibits are possible? 12 • 11 = 132 10 • 9 • 8 • 7 = 5,040 = 20P4 = 20 • 19 • 18 • 17 = 116,280

  16. Homework Assignment: Page 344 #1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, and 25

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