{"id":460,"date":"2019-07-15T22:45:17","date_gmt":"2019-07-15T22:45:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/combinations\/"},"modified":"2019-07-15T22:45:17","modified_gmt":"2019-07-15T22:45:17","slug":"combinations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/combinations\/","title":{"raw":"Combinations","rendered":"Combinations"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Find the number of combinations of n distinct choices.<\/li>\n<\/ul>\n<\/div>\n\n\nSo far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with <strong>combinations<\/strong>. A selection of [latex]r[\/latex] objects from a set of [latex]n[\/latex] objects where the order does not matter can be written as [latex]C\\left(n,r\\right)[\/latex]. Just as with permutations, [latex]\\text{C}\\left(n,r\\right)[\/latex] can also be written as [latex]{}_{n}{C}_{r}[\/latex]. In this case, the general formula is as follows.\n<p style=\"text-align: center\">[latex]\\text{C}\\left(n,r\\right)=\\dfrac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/p>\nAn earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are [latex]3!=3\\cdot 2\\cdot 1=6[\/latex] ways to order 3 paintings. There are [latex]\\frac{24}{6}[\/latex], or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are <em>not<\/em> selecting 1 painting. There are 4 paintings we could choose <em>not<\/em> to select, so there are 4 ways to select 3 of the 4 paintings.\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\nNote the similarity and difference between the formulas for permutations and combinations:\n\nPermutations (order matters), [latex]P(n, r)=\\dfrac{n!}{(n-r)!}[\/latex]\n\nCombinations (order does not matter), [latex]C(n, r)=\\dfrac{n!}{r!(n-r)!}[\/latex]\n\nThe formula for combinations is the formula for permutations with the number of ways to order [latex]r[\/latex] objects divided away from the result.\n\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Formula for Combinations of <em>n<\/em> Distinct Objects<\/h3>\nGiven [latex]n[\/latex] distinct objects, the number of ways to select [latex]r[\/latex] objects from the set is\n<p style=\"text-align: center\">[latex]\\text{C}\\left(n,r\\right)=\\dfrac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a number of options, determine the possible number of combinations.<\/h3>\n<ol>\n \t<li>Identify [latex]n[\/latex] from the given information.<\/li>\n \t<li>Identify [latex]r[\/latex] from the given information.<\/li>\n \t<li>Replace [latex]n[\/latex] and [latex]r[\/latex] in the formula with the given values.<\/li>\n \t<li>Evaluate.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Number of Combinations Using the Formula<\/h3>\nA fast food restaurant offers five side dish options. Your meal comes with two side dishes.\n<ol>\n \t<li>How many ways can you select your side dishes?<\/li>\n \t<li>How many ways can you select 3 side dishes?<\/li>\n<\/ol>\n[reveal-answer q=\"218844\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"218844\"]\n<ol>\n \t<li>We want to choose 2 side dishes from 5 options.\n[latex]\\text{C}\\left(5,2\\right)=\\dfrac{5!}{2!\\left(5 - 2\\right)!}=10[\/latex]<\/li>\n \t<li>We want to choose 3 side dishes from 5 options.\n[latex]\\text{C}\\left(5,3\\right)=\\dfrac{5!}{3!\\left(5 - 3\\right)!}=10[\/latex]<\/li>\n<\/ol>\n<h4>Analysis of the Solution<\/h4>\nWe can also use a graphing calculator to find combinations. Enter 5, then press [latex]{}_{n}{C}_{r}[\/latex], enter 3, and then press the equal sign. The [latex]{}_{n}{C}_{r}[\/latex], function may be located under the MATH menu with probability commands.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>Is it a coincidence that parts (a) and (b) in Example 4&nbsp;have the same answers?<\/h4>\n<em>No. When we choose r objects from n objects, we are <strong>not<\/strong> choosing [latex]\\left(n-r\\right)[\/latex] objects. Therefore, [latex]C\\left(n,r\\right)=C\\left(n,n-r\\right)[\/latex].<\/em>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nAn ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?\n\n[reveal-answer q=\"536738\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"536738\"]\n\n[latex]C\\left(10,3\\right)=120[\/latex]\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7156&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\n\n<\/div>\n<h2>Finding the Number of Subsets of a Set<\/h2>\nWe have looked only at combination problems in which we chose exactly [latex]r[\/latex] objects. In some problems, we want to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings. Any number of toppings can be ordered. How many different pizzas are possible?\n\nTo answer this question, we need to consider pizzas with any number of toppings. There is [latex]C\\left(5,0\\right)=1[\/latex] way to order a pizza with no toppings. There are [latex]C\\left(5,1\\right)=5[\/latex] ways to order a pizza with exactly one topping. If we continue this process, we get\n<p style=\"text-align: center\">[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>\nThere are 32 possible pizzas. This result is equal to [latex]{2}^{5}[\/latex].\n\nWe are presented with a sequence of choices. For each of the [latex]n[\/latex] objects we have two choices: include it in the subset or not. So for the whole subset we have made [latex]n[\/latex] choices, each with two options. So there are a total of [latex]2\\cdot 2\\cdot 2\\cdot \\dots \\cdot 2[\/latex] possible resulting subsets, all the way from the empty subset, which we obtain when we say \"no\" each time, to the original set itself, which we obtain when we say \"yes\" each time.\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\nDid you notice a pattern when you calculated the 32 possible pizzas long-hand?\n\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)=1+5+10+10+5+1=32[\/latex]\n\nThat is not a coincidence! We'll look more deeply at this phenomenon in the next section.\n\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Formula for the Number of Subsets of a Set<\/h3>\nA set containing <em>n<\/em> distinct objects has [latex]{2}^{n}[\/latex] subsets.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Number of Subsets of a Set<\/h3>\nA restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are there to order a potato?\n\n[reveal-answer q=\"961255\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"961255\"]\n\nWe are looking for the number of subsets of a set with 4 objects. Substitute [latex]n=4[\/latex] into the formula.\n<p style=\"text-align: center\">[latex]{2}^{n}={2}^{4} =16[\/latex]<\/p>\nThere are 16 possible ways to order a potato.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nA sundae bar at a wedding has 6 toppings to choose from. Any number of toppings can be chosen. How many different sundaes are possible?\n\n[reveal-answer q=\"719477\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"719477\"]\n\n64 sundaes\n\n[\/hidden-answer]\n\n[ohm_question]7175[\/ohm_question]\n\n<\/div>\n&nbsp;\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the number of combinations of n distinct choices.<\/li>\n<\/ul>\n<\/div>\n<p>So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with <strong>combinations<\/strong>. A selection of [latex]r[\/latex] objects from a set of [latex]n[\/latex] objects where the order does not matter can be written as [latex]C\\left(n,r\\right)[\/latex]. Just as with permutations, [latex]\\text{C}\\left(n,r\\right)[\/latex] can also be written as [latex]{}_{n}{C}_{r}[\/latex]. In this case, the general formula is as follows.<\/p>\n<p style=\"text-align: center\">[latex]\\text{C}\\left(n,r\\right)=\\dfrac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/p>\n<p>An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are [latex]3!=3\\cdot 2\\cdot 1=6[\/latex] ways to order 3 paintings. There are [latex]\\frac{24}{6}[\/latex], or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are <em>not<\/em> selecting 1 painting. There are 4 paintings we could choose <em>not<\/em> to select, so there are 4 ways to select 3 of the 4 paintings.<\/p>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>Note the similarity and difference between the formulas for permutations and combinations:<\/p>\n<p>Permutations (order matters), [latex]P(n, r)=\\dfrac{n!}{(n-r)!}[\/latex]<\/p>\n<p>Combinations (order does not matter), [latex]C(n, r)=\\dfrac{n!}{r!(n-r)!}[\/latex]<\/p>\n<p>The formula for combinations is the formula for permutations with the number of ways to order [latex]r[\/latex] objects divided away from the result.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Formula for Combinations of <em>n<\/em> Distinct Objects<\/h3>\n<p>Given [latex]n[\/latex] distinct objects, the number of ways to select [latex]r[\/latex] objects from the set is<\/p>\n<p style=\"text-align: center\">[latex]\\text{C}\\left(n,r\\right)=\\dfrac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a number of options, determine the possible number of combinations.<\/h3>\n<ol>\n<li>Identify [latex]n[\/latex] from the given information.<\/li>\n<li>Identify [latex]r[\/latex] from the given information.<\/li>\n<li>Replace [latex]n[\/latex] and [latex]r[\/latex] in the formula with the given values.<\/li>\n<li>Evaluate.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Number of Combinations Using the Formula<\/h3>\n<p>A fast food restaurant offers five side dish options. Your meal comes with two side dishes.<\/p>\n<ol>\n<li>How many ways can you select your side dishes?<\/li>\n<li>How many ways can you select 3 side dishes?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q218844\">Show Solution<\/span><\/p>\n<div id=\"q218844\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>We want to choose 2 side dishes from 5 options.<br \/>\n[latex]\\text{C}\\left(5,2\\right)=\\dfrac{5!}{2!\\left(5 - 2\\right)!}=10[\/latex]<\/li>\n<li>We want to choose 3 side dishes from 5 options.<br \/>\n[latex]\\text{C}\\left(5,3\\right)=\\dfrac{5!}{3!\\left(5 - 3\\right)!}=10[\/latex]<\/li>\n<\/ol>\n<h4>Analysis of the Solution<\/h4>\n<p>We can also use a graphing calculator to find combinations. Enter 5, then press [latex]{}_{n}{C}_{r}[\/latex], enter 3, and then press the equal sign. The [latex]{}_{n}{C}_{r}[\/latex], function may be located under the MATH menu with probability commands.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>Is it a coincidence that parts (a) and (b) in Example 4&nbsp;have the same answers?<\/h4>\n<p><em>No. When we choose r objects from n objects, we are <strong>not<\/strong> choosing [latex]\\left(n-r\\right)[\/latex] objects. Therefore, [latex]C\\left(n,r\\right)=C\\left(n,n-r\\right)[\/latex].<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q536738\">Show Solution<\/span><\/p>\n<div id=\"q536738\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]C\\left(10,3\\right)=120[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm7156\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7156&#38;theme=oea&#38;iframe_resize_id=ohm7156&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Finding the Number of Subsets of a Set<\/h2>\n<p>We have looked only at combination problems in which we chose exactly [latex]r[\/latex] objects. In some problems, we want to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings. Any number of toppings can be ordered. How many different pizzas are possible?<\/p>\n<p>To answer this question, we need to consider pizzas with any number of toppings. There is [latex]C\\left(5,0\\right)=1[\/latex] way to order a pizza with no toppings. There are [latex]C\\left(5,1\\right)=5[\/latex] ways to order a pizza with exactly one topping. If we continue this process, we get<\/p>\n<p style=\"text-align: center\">[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<p>There are 32 possible pizzas. This result is equal to [latex]{2}^{5}[\/latex].<\/p>\n<p>We are presented with a sequence of choices. For each of the [latex]n[\/latex] objects we have two choices: include it in the subset or not. So for the whole subset we have made [latex]n[\/latex] choices, each with two options. So there are a total of [latex]2\\cdot 2\\cdot 2\\cdot \\dots \\cdot 2[\/latex] possible resulting subsets, all the way from the empty subset, which we obtain when we say &#8220;no&#8221; each time, to the original set itself, which we obtain when we say &#8220;yes&#8221; each time.<\/p>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>Did you notice a pattern when you calculated the 32 possible pizzas long-hand?<\/p>\n<p>[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)=1+5+10+10+5+1=32[\/latex]<\/p>\n<p>That is not a coincidence! We&#8217;ll look more deeply at this phenomenon in the next section.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Formula for the Number of Subsets of a Set<\/h3>\n<p>A set containing <em>n<\/em> distinct objects has [latex]{2}^{n}[\/latex] subsets.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Number of Subsets of a Set<\/h3>\n<p>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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q961255\">Show Solution<\/span><\/p>\n<div id=\"q961255\" class=\"hidden-answer\" style=\"display: none\">\n<p>We are looking for the number of subsets of a set with 4 objects. Substitute [latex]n=4[\/latex] into the formula.<\/p>\n<p style=\"text-align: center\">[latex]{2}^{n}={2}^{4} =16[\/latex]<\/p>\n<p>There are 16 possible ways to order a potato.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A sundae bar at a wedding has 6 toppings to choose from. Any number of toppings can be chosen. How many different sundaes are possible?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q719477\">Show Solution<\/span><\/p>\n<div id=\"q719477\" class=\"hidden-answer\" style=\"display: none\">\n<p>64 sundaes<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm7175\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7175&theme=oea&iframe_resize_id=ohm7175&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-460\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 7175, 7156. <strong>Authored by<\/strong>: unknown, mb Lippman,David, mb Sousa,James. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Determine the Number of Ways 6 Books can be Selected from 9 Books (Combination). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/suCqU00Hh_s\">https:\/\/youtu.be\/suCqU00Hh_s<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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(Mathispower4u.com)\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/youtu.be\/suCqU00Hh_s\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"6b87d3b7-4e0e-45ed-b43d-29d78bb52cce","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-460","chapter","type-chapter","status-publish","hentry"],"part":453,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/460","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/460\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/parts\/453"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/460\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/media?parent=460"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=460"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/contributor?post=460"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/license?post=460"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}