{"id":296,"date":"2017-08-24T07:07:41","date_gmt":"2017-08-24T07:07:41","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=296"},"modified":"2017-08-31T07:01:51","modified_gmt":"2017-08-31T07:01:51","slug":"utilizing-combinations-and-permutations","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/utilizing-combinations-and-permutations\/","title":{"rendered":"Utilizing Combinations and Permutations"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/combinations-and-permutations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/probability-statistics-data-analysis\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/independent-and-dependent-events\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Utilizing Combinations and Permutations<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study how to use combinations and permutations to compute probability and<br \/>\nlearn useful simplifications created by using addition, multiplication, and complementation of known<br \/>\nprobabilities.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>The <strong><em>probability\u00a0<\/em><\/strong>of a favorable outcome from an event is a number between 0 and 1,\u00a0which measures the likelihood that the favorable outcome will\u00a0occur by using the formula:<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p1_clip_image003.gif\" width=\"265\" height=\"37\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0for the special cases where each outcome is equally likely.<\/li>\n<li>The <em><strong>sample space <\/strong><\/em>of\u00a0an event is the set consisting of all possible outcomes from the\u00a0event.<\/li>\n<li>The <em><strong>fundamental\u00a0counting principle <\/strong><\/em>states that the total number of \u00a0possible outcomes from a series of events is determined by\u00a0multiplying all the ways that each individual event can occur.<\/li>\n<li>The <em><strong>permutation\u00a0<\/strong><\/em>operation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p1_clip_image006.gif\" width=\"18\" height=\"16\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is used to count the elements of the sample space when orderin<span style=\"text-decoration: none;\">g\u00a0<\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">elements out of a set of <\/span><em><span style=\"text-decoration: none;\">n\u00a0<\/span><\/em><span style=\"text-decoration: none;\">elements and when order does matter. <\/span><\/li>\n<li><span style=\"text-decoration: none;\">The <\/span><strong><em><u>combination<\/u><\/em><\/strong><span style=\"text-decoration: none;\"> operation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p1_clip_image009.gif\" width=\"21\" height=\"16\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is used to count the ele<\/span>ments of the sample space when\u00a0group<span style=\"text-decoration: none;\">ing <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">elements out of a set of <\/span><em><span style=\"text-decoration: none;\">n\u00a0<\/span><\/em><span style=\"text-decoration: none;\">element<\/span>s and when order does not matter.<\/li>\n<\/ul>\n<section>\n<h3>How do permutations and combinations fit into determining\u00a0probability?<\/h3>\n<p>After defining probability, we took a little detour to talk\u00a0about permutations and combinations, which are really just\u00a0advanced counting techniques. We did this because probability is\u00a0simply a ratio of two numbers (favorable outcomes and total\u00a0outcomes) and we will need to use these two techniques to make a\u00a0quick and accurate count of these outcomes.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Suppose you have 10 different t-shirts, one of which is your\u00a0favorite. If you close your eyes and line up the t-shirts you will\u00a0wear for each day next week, what is the probability you will get\u00a0to wear your favorite t-shirt on Monday?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p2_clip_image003.gif\" width=\"9\" height=\"34\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p2_clip_image006.gif\" width=\"26\" height=\"39\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p2_clip_image009.gif\" width=\"29\" height=\"39\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p2_clip_image012.gif\" width=\"16\" height=\"34\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D. In this example, <span style=\"text-decoration: none;\">the\u00a0outcome <\/span><em><span style=\"text-decoration: none;\">E\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= &#8220;I will pick my favorite t-shirt for Monday.&#8221; Notice\u00a0that because we are arranging the t-shirts in a specific order,\u00a0we should use perm<\/span>utations.<\/p>\n<p>First, calculate the total number of possible ways that your\u00a0t-shirts for next week can be ordered. We use\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p2_clip_image015.gif\" width=\"22\" height=\"16\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>\u00a0to calculate the total number of ways to order 7 t-shirts out of\u00a010.<\/p>\n<p>Now we will use the fundamental counting principle to count\u00a0the number of favorable outcomes for this event. List the days of\u00a0the week.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p2_clip_image018.gif\" width=\"281\" height=\"13\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>For a favorable outcome, you must choose your favorite t-shirt\u00a0for Monday. Thus there is only one favorable choice for Monday.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p2_clip_image021.gif\" width=\"201\" height=\"24\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>With one t-shirt spoken for, we have 9 options for Sunday\u2019s\u00a0t-shirt.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p2_clip_image024.gif\" width=\"208\" height=\"24\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Then there are 8 options remaining for Tuesday\u2019s\u00a0t-shirt.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p2_clip_image027.gif\" width=\"208\" height=\"24\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>This leaves 7 options for Wednesday\u2019s t-shirt, and so\u00a0on.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p2_clip_image030.gif\" width=\"207\" height=\"24\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Ultimately, we get<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/s3_p2_html_7edda840.gif\" width=\"636\" height=\"63\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>We could rethink this question in the following way to justify the\u00a0answer:<\/p>\n<p>What if today is Monday and we want to know the probability of picking\u00a0our favorite t-shirt today? Probabilistically speaking, these two situations are the same. There is only\u00a0one favorable outcome out of 10 possible outcomes. This gives us <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p2_clip_image036.gif\" width=\"66\" height=\"34\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p><span style=\"text-decoration: none;\">In the card game 5-card\u00a0stud, each player is dealt five cards per hand. Suppose you are\u00a0playing a game of 5-card stud with 4 friends. After playing the\u00a0first round, the cards are thoroughly shuffled and dealt again.\u00a0What is the probability that you will be dealt the <\/span><em><span style=\"text-decoration: none;\">exact\u00a0<\/span><\/em><span style=\"text-decoration: none;\">hand you started with in the first round? (There are 52 cards in a\u00a0dec<\/span>k.)<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p3_clip_image003.gif\" width=\"29\" height=\"40\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p3_clip_image006.gif\" width=\"29\" height=\"39\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p3_clip_image009.gif\" width=\"29\" height=\"40\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p3_clip_image012.gif\" width=\"25\" height=\"39\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is B. Since it doesn\u2019t matter what\u00a0order you are dealt the cards in your hand, you should\u00a0immediately think of combinations. The cards are being dealt to\u00a0you, so the dealer is choosing 5 cards out of a deck of 52. You\u00a0want to get the exact hand you had in the first round, so there\u00a0is only 1 favorable outcome. Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p3_clip_image015.gif\" width=\"80\" height=\"39\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>Use combinations here because the order in which you receive the\u00a0cards does not\u00a0change the hand you are dealt.<\/p>\n<\/div>\n<h4>How can probabilities involving two events be simplified?<\/h4>\n<p><span style=\"text-decoration: none;\">Suppose there are two\u00a0events, each with their own favorable outcomes\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image003.gif\" width=\"60\" height=\"16\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and each with their own probabilities\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image006.gif\" width=\"105\" height=\"16\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0We can ask what is the probability of <\/span><em><span style=\"text-decoration: none;\">both\u00a0<\/span><\/em><span style=\"text-decoration: none;\">favorable outcomes occurring, i.e.,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image009.gif\" width=\"89\" height=\"23\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0or we can ask what is the probability of <\/span><em><span style=\"text-decoration: none;\">at\u00a0least one<\/span><\/em><span style=\"text-decoration: none;\"> of the\u00a0favorable outcomes occurring, i.e.,<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image010.gif\" width=\"41\" height=\"24\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/>. We can even ask what the probability is of a favorable outcome\u00a0<\/span><em><span style=\"text-decoration: none;\">not <\/span><\/em><span style=\"text-decoration: none;\">occurring,\u00a0i.e.,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image012.gif\" width=\"62\" height=\"18\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0In all of these situations, there are simple formulas defined in\u00a0terms of the known probabilities. <\/span><\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image015.gif\" width=\"194\" height=\"23\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image018.gif\" width=\"284\" height=\"24\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image021.gif\" width=\"137\" height=\"18\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<\/div>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p><span style=\"text-decoration: none;\">All probabilities are less than or equal to one, so the product of any two is never larger than either of the two probabilities considered individually. Because many probabilities are less than one and we are evaluating the likelihood of <\/span><em><span style=\"text-decoration: none;\">both<\/span><\/em><span style=\"text-decoration: none;\"> favorable outcomes occurring, we would often expect the final probability to be <\/span><em><span style=\"text-decoration: none;\">smaller<\/span><\/em><span style=\"text-decoration: none;\"> than the individual probabilities. <\/span><\/p>\n<p class=\"notebox_text\" align=\"CENTER\"><span style=\"text-decoration: none;\">Adding two probabilities, on the other hand, <\/span><em><span style=\"text-decoration: none;\">increases<\/span><\/em><span style=\"text-decoration: none;\"> the total probability. Because we are evaluating the likelihood of <\/span><em><span style=\"text-decoration: none;\">either<\/span><\/em><span style=\"text-decoration: none;\"> favorable outcome occurring, we would expect the final likelihood to be greater. <\/span><\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p><span style=\"text-decoration: none;\">Suppose we pick a card at\u00a0random from a deck of 52 cards while rolling a fair six-sided die.\u00a0What is the probability of getting a <\/span><em><span style=\"text-decoration: none;\">red\u00a0face card<\/span><\/em><span style=\"text-decoration: none;\"> from the\u00a0deck and a <\/span><em><span style=\"text-decoration: none;\">number\u00a0less than 5<\/span><\/em><span style=\"text-decoration: none;\"> on the\u00a0die? Remember that there are 12 face cards in a deck, 3 in each\u00a0suit. <\/span><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image024.gif\" width=\"9\" height=\"34\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image027.gif\" width=\"17\" height=\"34\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image030.gif\" width=\"17\" height=\"34\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p3_clip_image016.gif\" width=\"21\" height=\"41\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D. Here\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image036.gif\" width=\"13\" height=\"15\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0= \u201cchoosing a red face card\u201d and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image039.gif\" width=\"14\" height=\"15\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>=\u00a0\u201cdie lands on a number less than 5.\u201d Because you are\u00a0looking for the probability of two independent events, use\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image042.gif\" width=\"191\" height=\"18\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0to make your calculations.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3.p4.001.gif\" width=\"110\" height=\"41\" align=\"absmiddle\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3.p4.002.gif\" width=\"96\" height=\"41\" align=\"absmiddle\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3.p4.003.gif\" width=\"273\" height=\"41\" \/><\/p>\n<\/div>\n<\/section>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/combinations-and-permutations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/probability-statistics-data-analysis\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/independent-and-dependent-events\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Utilizing Combinations and Permutations Objective In this lesson, you will study how to use combinations and permutations to compute probability and learn useful simplifications created by using addition, multiplication, and complementation of known probabilities. Previously Covered: The probability\u00a0of a favorable outcome from an event is a number between 0 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-296","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/296","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=296"}],"version-history":[{"count":7,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/296\/revisions"}],"predecessor-version":[{"id":632,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/296\/revisions\/632"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=296"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}