{"id":1748,"date":"2025-07-28T19:01:56","date_gmt":"2025-07-28T19:01:56","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1748"},"modified":"2026-03-26T17:03:30","modified_gmt":"2026-03-26T17:03:30","slug":"counting-principles-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/counting-principles-learn-it-5\/","title":{"raw":"Counting Principles: Learn It 5","rendered":"Counting Principles: Learn It 5"},"content":{"raw":"

Finding the Number of Subsets of a Set<\/h2>\r\nWe have looked only at combination problems in which we chose exactly objects. In some problems, we want to consider choosing every possible number of objects.\r\n\r\n
\r\n
\r\n
\r\n
\r\n\r\nA pizza restaurant that offers [latex]5[\/latex] toppings. Any number of toppings can be ordered. How many different pizzas are possible?\r\n\r\nTo answer this question, we need to consider pizzas with any number of toppings.\r\n
    \r\n \t
  • There is [latex]C\\left(5,0\\right)=1[\/latex] way to order a pizza with no toppings.<\/li>\r\n \t
  • There are [latex]C\\left(5,1\\right)=5[\/latex] ways to order a pizza with exactly one topping.<\/li>\r\n \t
  • If we continue this process, we get:<\/li>\r\n<\/ul>\r\n

    [latex]C\\left(5,0\\right)+C\\left(5,1\\right)+C\\left(5,2\\right)+C\\left(5,3\\right)+C\\left(5,4\\right)+C\\left(5,5\\right)=32[\/latex]<\/p>\r\nThere are [latex]32[\/latex] possible pizzas.\r\n\r\nThis result [latex]32[\/latex] is equal to [latex]{2}^{5}[\/latex] possible pizzas, which offers [latex]5[\/latex] toppings. Coincidence? Definitely not!\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>

    \r\n

    formula for the number of subsets of a set<\/h3>\r\nA set containing [latex]n[\/latex] distinct objects has [latex]{2}^{n}[\/latex] subsets.\r\n\r\n<\/section>
    A restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are there to order a potato?[reveal-answer q=\"961255\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"961255\"]We are looking for the number of subsets of a set with [latex]4[\/latex] objects. Substitute [latex]n=4[\/latex] into the formula.\r\n

    [latex]{2}^{n}={2}^{4} =16[\/latex]<\/p>\r\nThere are [latex]16[\/latex] possible ways to order a potato.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]321997[\/ohm_question]<\/section>","rendered":"

    Finding the Number of Subsets of a Set<\/h2>\n

    We have looked only at combination problems in which we chose exactly objects. In some problems, we want to consider choosing every possible number of objects.<\/p>\n

    \n
    \n
    \n
    \n

    A pizza restaurant that offers [latex]5[\/latex] toppings. Any number of toppings can be ordered. How many different pizzas are possible?<\/p>\n

    To answer this question, we need to consider pizzas with any number of toppings.<\/p>\n

      \n
    • There is [latex]C\\left(5,0\\right)=1[\/latex] way to order a pizza with no toppings.<\/li>\n
    • There are [latex]C\\left(5,1\\right)=5[\/latex] ways to order a pizza with exactly one topping.<\/li>\n
    • If we continue this process, we get:<\/li>\n<\/ul>\n

      [latex]C\\left(5,0\\right)+C\\left(5,1\\right)+C\\left(5,2\\right)+C\\left(5,3\\right)+C\\left(5,4\\right)+C\\left(5,5\\right)=32[\/latex]<\/p>\n

      There are [latex]32[\/latex] possible pizzas.<\/p>\n

      This result [latex]32[\/latex] is equal to [latex]{2}^{5}[\/latex] possible pizzas, which offers [latex]5[\/latex] toppings. Coincidence? Definitely not!<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n

      \n

      formula for the number of subsets of a set<\/h3>\n

      A set containing [latex]n[\/latex] distinct objects has [latex]{2}^{n}[\/latex] subsets.<\/p>\n<\/section>\n

      A restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are there to order a potato?<\/p>\n