{"id":161,"date":"2025-02-13T22:44:28","date_gmt":"2025-02-13T22:44:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/counting-principles\/"},"modified":"2026-03-26T16:49:16","modified_gmt":"2026-03-26T16:49:16","slug":"counting-principles","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/counting-principles\/","title":{"raw":"Counting Principles: Learn It 1","rendered":"Counting Principles: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Solve counting problems using the Addition and Multiplication Principles.<\/li>\r\n \t<li style=\"font-weight: 400;\">Solve counting problems using permutations and combinations.<\/li>\r\n \t<li style=\"font-weight: 400;\">Solve counting problems using permutations involving n non-distinct objects.<\/li>\r\n<\/ul>\r\n<\/section>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<h2>Using the Addition Principle<\/h2>\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 <strong>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<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03231533\/CNX_Precalc_Figure_11_05_001n2.jpg\" alt=\"The addition of 3 iPods and 4 iPhones.\" width=\"487\" height=\"358\" \/>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>addition principle<\/h3>\r\nThe <strong>Addition Principle<\/strong> states that if one event can occur in [latex]A[\/latex] ways ([latex]A[\/latex] outcomes) and a second event can occur in [latex]B[\/latex] ways ([latex]B[\/latex] outcomes) and both events <em>cannot occur at the same time<\/em> ([latex]A[\/latex] and [latex]B[\/latex] disjoints), then there are [latex]A B[\/latex] ways ([latex]A B[\/latex] outcomes) for the first event <em>OR<\/em> the second event to occur.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A student is shopping for a new computer. He is deciding among [latex]3[\/latex] desktop computers and [latex]4[\/latex] laptop computers. What is the total number of computer options?[reveal-answer q=\"856363\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"856363\"][latex]7[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321988[\/ohm_question]<\/section><section aria-label=\"Try It\">\r\n<h2>Using the Multiplication Principle<\/h2>\r\nThe <strong>Multiplication Principle<\/strong> applies when we are making more than one selection.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Suppose we are choosing an appetizer, an entr\u00e9e, and a dessert. If there are [latex]2[\/latex] appetizer options, [latex]3[\/latex] entr\u00e9e options, and [latex]2[\/latex] dessert options on a fixed-price dinner menu, there are a total of [latex]12[\/latex] possible choices of one each as shown in the tree diagram.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03231541\/CNX_Precalc_Figure_11_05_0032.jpg\" alt=\"A tree diagram of the different menu combinations.\" width=\"975\" height=\"287\" \/>\r\n<ol>\r\n \t<li>soup, chicken, cake<\/li>\r\n \t<li>soup, chicken, pudding<\/li>\r\n \t<li>soup, fish, cake<\/li>\r\n \t<li>soup, fish, pudding<\/li>\r\n \t<li>soup, steak, cake<\/li>\r\n \t<li>soup, steak, pudding<\/li>\r\n \t<li>salad, chicken, cake<\/li>\r\n \t<li>salad, chicken, pudding<\/li>\r\n \t<li>salad, fish, cake<\/li>\r\n \t<li>salad, fish, pudding<\/li>\r\n \t<li>salad, steak, cake<\/li>\r\n \t<li>salad, steak, pudding<\/li>\r\n<\/ol>\r\nWe can also find the total number of possible dinners by multiplying.\r\n\r\n[latex]\\begin{array}{cccccc} \\# \\text{ of appetizer options} &amp; \\times &amp; \\# \\text{ of entree options} &amp; \\times &amp; \\# \\text{ of dessert options} \\\\[6pt] 2 &amp; \\times &amp; 3 &amp; \\times &amp; 2 &amp; = 12 \\end{array}[\/latex]\r\n\r\nThus, <strong>there are [latex]12[\/latex] possible dinner choices<\/strong> simply by applying the Multiplication Principle.\r\n\r\n<\/section><\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>multiplication principle<\/h3>\r\nThe <strong>Multiplication Principle<\/strong> states that if one event can occur in [latex]A[\/latex] ways ([latex]A[\/latex] outcomes) and a second event can occur in [latex]B[\/latex] ways ([latex]B[\/latex] outcomes) after the first event has occurred then the two events can occur in [latex]A \\cdot B[\/latex] ways.\r\n\r\n&nbsp;\r\n\r\nThis is also known as the <strong>Fundamental Counting Principle<\/strong>.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Diane packed [latex]2[\/latex] skirts, [latex]4[\/latex] 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.[reveal-answer q=\"912778\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"912778\"]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.\r\n[latex]\\begin{array}{cccccc} \\# \\text{ of skirt options} &amp; \\times &amp; \\# \\text{ of blouse options} &amp; \\times &amp; \\# \\text{ of sweater options} \\\\[6pt] 2 &amp; \\times &amp; 4 &amp; \\times &amp; 2 &amp; = 16 \\end{array}[\/latex]\r\nThere are [latex]16[\/latex] possible outfits.\r\n[\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321989[\/ohm_question]<\/section><\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Solve counting problems using the Addition and Multiplication Principles.<\/li>\n<li style=\"font-weight: 400;\">Solve counting problems using permutations and combinations.<\/li>\n<li style=\"font-weight: 400;\">Solve counting problems using permutations involving n non-distinct objects.<\/li>\n<\/ul>\n<\/section>\n<p>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!<\/p>\n<p>We 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.<\/p>\n<h2>Using the Addition Principle<\/h2>\n<p>The company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet models and 5 supported smartphone models. The <strong>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.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03231533\/CNX_Precalc_Figure_11_05_001n2.jpg\" alt=\"The addition of 3 iPods and 4 iPhones.\" width=\"487\" height=\"358\" \/><\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>addition principle<\/h3>\n<p>The <strong>Addition Principle<\/strong> states that if one event can occur in [latex]A[\/latex] ways ([latex]A[\/latex] outcomes) and a second event can occur in [latex]B[\/latex] ways ([latex]B[\/latex] outcomes) and both events <em>cannot occur at the same time<\/em> ([latex]A[\/latex] and [latex]B[\/latex] disjoints), then there are [latex]A B[\/latex] ways ([latex]A B[\/latex] outcomes) for the first event <em>OR<\/em> the second event to occur.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A student is shopping for a new computer. He is deciding among [latex]3[\/latex] desktop computers and [latex]4[\/latex] laptop computers. What is the total number of computer options?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q856363\">Show Solution<\/button><\/p>\n<div id=\"q856363\" class=\"hidden-answer\" style=\"display: none\">[latex]7[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321988\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321988&theme=lumen&iframe_resize_id=ohm321988&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<h2>Using the Multiplication Principle<\/h2>\n<p>The <strong>Multiplication Principle<\/strong> applies when we are making more than one selection.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Suppose we are choosing an appetizer, an entr\u00e9e, and a dessert. If there are [latex]2[\/latex] appetizer options, [latex]3[\/latex] entr\u00e9e options, and [latex]2[\/latex] dessert options on a fixed-price dinner menu, there are a total of [latex]12[\/latex] possible choices of one each as shown in the tree diagram.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03231541\/CNX_Precalc_Figure_11_05_0032.jpg\" alt=\"A tree diagram of the different menu combinations.\" width=\"975\" height=\"287\" \/><\/p>\n<ol>\n<li>soup, chicken, cake<\/li>\n<li>soup, chicken, pudding<\/li>\n<li>soup, fish, cake<\/li>\n<li>soup, fish, pudding<\/li>\n<li>soup, steak, cake<\/li>\n<li>soup, steak, pudding<\/li>\n<li>salad, chicken, cake<\/li>\n<li>salad, chicken, pudding<\/li>\n<li>salad, fish, cake<\/li>\n<li>salad, fish, pudding<\/li>\n<li>salad, steak, cake<\/li>\n<li>salad, steak, pudding<\/li>\n<\/ol>\n<p>We can also find the total number of possible dinners by multiplying.<\/p>\n<p>[latex]\\begin{array}{cccccc} \\# \\text{ of appetizer options} & \\times & \\# \\text{ of entree options} & \\times & \\# \\text{ of dessert options} \\\\[6pt] 2 & \\times & 3 & \\times & 2 & = 12 \\end{array}[\/latex]<\/p>\n<p>Thus, <strong>there are [latex]12[\/latex] possible dinner choices<\/strong> simply by applying the Multiplication Principle.<\/p>\n<\/section>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>multiplication principle<\/h3>\n<p>The <strong>Multiplication Principle<\/strong> states that if one event can occur in [latex]A[\/latex] ways ([latex]A[\/latex] outcomes) and a second event can occur in [latex]B[\/latex] ways ([latex]B[\/latex] outcomes) after the first event has occurred then the two events can occur in [latex]A \\cdot B[\/latex] ways.<\/p>\n<p>&nbsp;<\/p>\n<p>This is also known as the <strong>Fundamental Counting Principle<\/strong>.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Diane packed [latex]2[\/latex] skirts, [latex]4[\/latex] 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 class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q912778\">Show Solution<\/button><\/p>\n<div id=\"q912778\" class=\"hidden-answer\" style=\"display: none\">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.<br \/>\n[latex]\\begin{array}{cccccc} \\# \\text{ of skirt options} & \\times & \\# \\text{ of blouse options} & \\times & \\# \\text{ of sweater options} \\\\[6pt] 2 & \\times & 4 & \\times & 2 & = 16 \\end{array}[\/latex]<br \/>\nThere are [latex]16[\/latex] possible outfits.\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321989\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321989&theme=lumen&iframe_resize_id=ohm321989&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n","protected":false},"author":6,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":513,"module-header":"learn_it","content_attributions":[{"type":"original","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/161"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":16,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/161\/revisions"}],"predecessor-version":[{"id":6043,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/161\/revisions\/6043"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/513"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/161\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=161"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=161"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=161"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=161"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}