{"id":5252,"date":"2018-05-12T18:30:48","date_gmt":"2018-05-12T18:30:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/coreq-collegealgebra\/?post_type=chapter&p=5252"},"modified":"2018-05-21T20:21:14","modified_gmt":"2018-05-21T20:21:14","slug":"counting-principles","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/counting-principles\/","title":{"raw":"Counting Principles*","rendered":"Counting Principles*"},"content":{"raw":"A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. The customer can choose the order of the images and the letters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. Counting the possibilities is challenging!\r\n\r\nWe encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. We will examine this type of mathematics in this section.\r\n

Permutations<\/h2>\r\n

Using the Addition Principle<\/h3>\r\nThe company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet models and 5 supported smartphone models. The Addition Principle<\/strong> tells us that we can add the number of tablet options to the number of smartphone options to find the total number of options. By the Addition Principle, there are 8 total options.\r\n\r\n\"The\r\n
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A General Note: The Addition Principle<\/h3>\r\nAccording to the Addition Principle<\/strong>, if one event can occur in [latex]m[\/latex] ways and a second event with no common outcomes can occur in [latex]n[\/latex] ways, then the first or<\/em> second event can occur in [latex]m+n[\/latex] ways.\r\n\r\n<\/div>\r\n
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Example: Using the Addition Principle<\/h3>\r\nThere are 2 vegetarian entr\u00e9e options and 5 meat entr\u00e9e options on a dinner menu. What is the total number of entr\u00e9e options?\r\n\r\n[reveal-answer q=\"477104\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"477104\"]\r\n\r\nWe can add the number of vegetarian options to the number of meat options to find the total number of entr\u00e9e options.\r\n\r\n\"The\r\n\r\nThere are 7 total options.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
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Try It<\/h3>\r\nA student is shopping for a new computer. He is deciding among 3 desktop computers and 4 laptop computers. What is the total number of computer options?\r\n\r\n[reveal-answer q=\"856363\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"856363\"]7[\/hidden-answer]\r\n\r\n<\/div>\r\n

Using the Multiplication Principle<\/h3>\r\nThe Multiplication Principle<\/strong> applies when we are making more than one selection. Suppose we are choosing an appetizer, an entr\u00e9e, and a dessert. If there are 2 appetizer options, 3 entr\u00e9e options, and 2 dessert options on a fixed-price dinner menu, there are a total of 12 possible choices of one each as shown in the tree diagram.\r\n\r\n\"A\r\n\r\nThe possible choices are:\r\n
    \r\n \t
  1. soup, chicken, cake<\/li>\r\n \t
  2. soup, chicken, pudding<\/li>\r\n \t
  3. soup, fish, cake<\/li>\r\n \t
  4. soup, fish, pudding<\/li>\r\n \t
  5. soup, steak, cake<\/li>\r\n \t
  6. soup, steak, pudding<\/li>\r\n \t
  7. salad, chicken, cake<\/li>\r\n \t
  8. salad, chicken, pudding<\/li>\r\n \t
  9. salad, fish, cake<\/li>\r\n \t
  10. salad, fish, pudding<\/li>\r\n \t
  11. salad, steak, cake<\/li>\r\n \t
  12. salad, steak, pudding<\/li>\r\n<\/ol>\r\nWe can also find the total number of possible dinners by multiplying.\r\n\r\nWe could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.\r\n\"Number<\/a>\r\n
    \r\n

    A General Note: The Multiplication Principle<\/h3>\r\nAccording to the Multiplication Principle<\/strong>, if one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\\times n[\/latex] ways. This is also known as the Fundamental Counting Principle<\/strong>.\r\n\r\n<\/div>\r\n
    \r\n

    Example: Using the Multiplication Principle<\/h3>\r\nDiane packed 2 skirts, 4 blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for each outfit and decide whether to wear the sweater. Use the Multiplication Principle to find the total number of possible outfits.\r\n\r\n[reveal-answer q=\"912778\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"912778\"]\r\n\r\nTo find the total number of outfits, find the product of the number of skirt options, the number of blouse options, and the number of sweater options.\r\n\r\n\"The\r\n\r\nThere are 16 possible outfits.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
    \r\n

    Try It<\/h3>\r\nA restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are 3 types of breakfast sandwiches, 4 side dish options, and 5 beverage choices. Find the total number of possible breakfast specials.\r\n\r\n[reveal-answer q=\"559695\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"559695\"]There are 60 possible breakfast specials.[\/hidden-answer]\r\n\r\n<\/div>\r\nThe Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation<\/strong>.\r\n

    Finding the Number of Permutations of n<\/em> Distinct Objects Using the Multiplication Principle<\/h3>\r\nTo solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.\r\n\r\n\"\"\r\n\r\nThere are four options for the first place, so we write a 4 on the first line.\r\n\r\n\"Four\r\n\r\nAfter the first place has been filled, there are three options for the second place so we write a 3 on the second line.\r\n\r\n\"Four\r\n\r\nAfter the second place has been filled, there are two options for the third place so we write a 2 on the third line. Finally, we find the product.\r\n\r\n\"\"\r\n\r\nThere are 24 possible permutations of the paintings.\r\n
    \r\n

    How To: Given [latex]n[\/latex] distinct options, determine how many permutations there are.<\/h3>\r\n
      \r\n \t
    1. Determine how many options there are for the first situation.<\/li>\r\n \t
    2. Determine how many options are left for the second situation.<\/li>\r\n \t
    3. Continue until all of the spots are filled.<\/li>\r\n \t
    4. Multiply the numbers together.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Example: Finding the Number of Permutations Using the Multiplication Principle<\/h3>\r\nAt a swimming competition, nine swimmers compete in a race.\r\n
        \r\n \t
      1. How many ways can they place first, second, and third?<\/li>\r\n \t
      2. How many ways can they place first, second, and third if a swimmer named Ariel wins first place? (Assume there is only one contestant named Ariel.)<\/li>\r\n \t
      3. How many ways can all nine swimmers line up for a photo?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"268922\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"268922\"]\r\n
          \r\n \t
        1. Draw lines for each place.\"\"There are 9 options for first place. Once someone has won first place, there are 8 remaining options for second place. Once first and second place have been won, there are 7 remaining options for third place.\"\"Multiply to find that there are 504 ways for the swimmers to place.<\/li>\r\n \t
        2. Draw lines for describing each place.\"\"We know Ariel must win first place, so there is only 1 option for first place. There are 8 remaining options for second place, and then 7 remaining options for third place.\"\"Multiply to find that there are 56 ways for the swimmers to place if Ariel wins first.<\/li>\r\n \t
        3. Draw lines for describing each place in the photo.\"\"There are 9 choices for the first spot, then 8 for the second, 7 for the third, 6 for the fourth, and so on until only 1 person remains for the last spot.\"\"There are 362,880 possible permutations for the swimmers to line up.<\/li>\r\n<\/ol>\r\n

          Analysis of the Solution<\/h4>\r\nNote that in part c, we found there were 9! ways for 9 people to line up. The number of permutations of [latex]n[\/latex] distinct objects can always be found by [latex]n![\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
          \r\n

          Try It<\/h3>\r\nA family of five is having portraits taken. Use the Multiplication Principle to find the following.\r\nHow many ways can the family line up for the portrait?\r\n\r\n[reveal-answer q=\"847892\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"847892\"]120[\/hidden-answer]\r\n\r\nHow many ways can the photographer line up 3 family members?\r\n\r\n[reveal-answer q=\"416580\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"416580\"]60[\/hidden-answer]\r\n\r\nHow many ways can the family line up for the portrait if the parents are required to stand on each end?\r\n\r\n[reveal-answer q=\"518944\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"518944\"]12[\/hidden-answer]\r\n\r\n