{"id":2085,"date":"2015-11-12T18:30:41","date_gmt":"2015-11-12T18:30:41","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=2085"},"modified":"2015-11-12T18:30:41","modified_gmt":"2015-11-12T18:30:41","slug":"using-the-addition-and-multiplication-principles","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/using-the-addition-and-multiplication-principles\/","title":{"raw":"Using the Addition and Multiplication Principles","rendered":"Using the Addition and Multiplication Principles"},"content":{"raw":"

The 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, as we can see in Figure 1.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"TheFigure 1<\/b>[\/caption]\n\n<\/p>

\n

A General Note: The Addition Principle<\/h3>\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.\n\n<\/div>\n
\n

Example 1: Using the Addition Principle<\/h3>\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?\n\n<\/div>\n
\n

Solution<\/h3>\nWe can add the number of vegetarian options to the number of meat options to find the total number of entr\u00e9e options.\n\n\"The\n\nThere are 7 total options.\n\n<\/div>\n
\n

Try It 1<\/h3>\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?\n\nSolution<\/a>\n\n<\/div>\n

Using the Multiplication Principle<\/h2>\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 in Figure 2.\n\n\"A\n\nThe possible choices are:\n
  1. soup, chicken, cake<\/li>\n\t
  2. soup, chicken, pudding<\/li>\n\t
  3. soup, fish, cake<\/li>\n\t
  4. soup, fish, pudding<\/li>\n\t
  5. soup, steak, cake<\/li>\n\t
  6. soup, steak, pudding<\/li>\n\t
  7. salad, chicken, cake<\/li>\n\t
  8. salad, chicken, pudding<\/li>\n\t
  9. salad, fish, cake<\/li>\n\t
  10. salad, fish, pudding<\/li>\n\t
  11. salad, steak, cake<\/li>\n\t
  12. salad, steak, pudding<\/li>\n<\/ol>\nWe can also find the total number of possible dinners by multiplying.\n\nWe could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.\n
    \"Number<\/a><\/div>\n
    \n

    A General Note: The Multiplication Principle<\/h3>\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>.\n\n<\/div>\n
    \n

    Example 1: Using the Multiplication Principle<\/h3>\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.\n\n<\/div>\n
    \n

    Solution<\/h3>\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.\n\n\"The\n\nThere are 16 possible outfits.\n\n<\/div>\n
    \n

    Try It 2<\/h3>\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.\n\nSolution<\/a>\n\n<\/div>","rendered":"

    The 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, as we can see in Figure 1.<\/p>\n

    \"The<\/p>\n

    Figure 1<\/b><\/p>\n<\/div>\n

    \n

    A General Note: The Addition Principle<\/h3>\n

    According 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.<\/p>\n<\/div>\n

    \n

    Example 1: Using the Addition Principle<\/h3>\n

    There 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?<\/p>\n<\/div>\n

    \n

    Solution<\/h3>\n

    We can add the number of vegetarian options to the number of meat options to find the total number of entr\u00e9e options.<\/p>\n

    \"The<\/p>\n

    There are 7 total options.<\/p>\n<\/div>\n

    \n

    Try It 1<\/h3>\n

    A 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?<\/p>\n

    Solution<\/a><\/p>\n<\/div>\n

    Using the Multiplication Principle<\/h2>\n

    The 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 in Figure 2.<\/p>\n

    \"A<\/p>\n

    The possible choices are:<\/p>\n

      \n
    1. soup, chicken, cake<\/li>\n
    2. soup, chicken, pudding<\/li>\n
    3. soup, fish, cake<\/li>\n
    4. soup, fish, pudding<\/li>\n
    5. soup, steak, cake<\/li>\n
    6. soup, steak, pudding<\/li>\n
    7. salad, chicken, cake<\/li>\n
    8. salad, chicken, pudding<\/li>\n
    9. salad, fish, cake<\/li>\n
    10. salad, fish, pudding<\/li>\n
    11. salad, steak, cake<\/li>\n
    12. salad, steak, pudding<\/li>\n<\/ol>\n

      We can also find the total number of possible dinners by multiplying.<\/p>\n

      We could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.<\/p>\n

      \"Number<\/a><\/div>\n
      \n

      A General Note: The Multiplication Principle<\/h3>\n

      According 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>.<\/p>\n<\/div>\n

      \n

      Example 1: Using the Multiplication Principle<\/h3>\n

      Diane 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.<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      To 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.<\/p>\n

      \"The<\/p>\n

      There are 16 possible outfits.<\/p>\n<\/div>\n

      \n

      Try It 2<\/h3>\n

      A 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.<\/p>\n

      Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

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