{"id":295,"date":"2017-08-24T07:07:18","date_gmt":"2017-08-24T07:07:18","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=295"},"modified":"2017-08-31T06:53:46","modified_gmt":"2017-08-31T06:53:46","slug":"combinations-and-permutations","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/combinations-and-permutations\/","title":{"rendered":"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\/defining-probability\">\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\/utilizing-combinations-and-permutations\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Combinations and Permutations<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study how to use combinations and permutations.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li><span style=\"text-decoration: none;\">The <\/span><strong><em><span style=\"text-decoration: none;\">!\u00a0<\/span><\/em><\/strong><span style=\"text-decoration: none;\">operation is called <\/span><strong><em><span style=\"text-decoration: none;\">factorial.<\/span><br \/>\n<\/em><\/strong><\/li>\n<\/ul>\n<section>\n<h3>Combinations and Permutations<\/h3>\n<p>Two other useful techniques that aid in counting are\u00a0<abbr title=\"a method used to count the number of ways you can arrange r elements out of n elements. It is given by the formula, c. \">permutations<\/abbr> and <abbr title=\"counts the number of ways to choose r elements out of a collection of n elements. It is given by the formula, MathType.\">combinations<\/abbr>.<\/p>\n<p>In the earlier example, we wanted to find the ways that 7 books\u00a0could be arranged in 7 spaces on a bookshelf. What if we had only\u00a05 spaces instead of 7 in which to place the 7 books?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image003.gif\" width=\"159\" height=\"11\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>We still have 7 choices for the first slot.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image006.gif\" width=\"161\" height=\"24\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>There are 6 choices for the second slot.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image009.gif\" width=\"161\" height=\"24\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>And so on.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image012.gif\" width=\"161\" height=\"24\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>There are\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image015.gif\" width=\"124\" height=\"13\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>\u00a0possibilities.<\/p>\n<p><span style=\"text-decoration: none;\">In this example, the order\u00a0of the books <\/span><em><span style=\"text-decoration: none;\">does\u00a0<\/span><\/em><span style=\"text-decoration: none;\">matter. If we\u00a0changed the order, we would change the element of the <\/span><strong><em><span style=\"text-decoration: none;\">sample\u00a0space<\/span><\/em><span style=\"text-decoration: none;\">, <\/span><\/strong><span style=\"text-decoration: none;\">which\u00a0is the set of all possible outcomes<\/span><strong><em><span style=\"text-decoration: none;\">.\u00a0<\/span><\/em><\/strong><span style=\"text-decoration: none;\">In this situation, we can simplify the computation by using a\u00a0permutation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image018.gif\" width=\"41\" height=\"14\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>.<br \/>\n<\/span><\/p>\n<p>The formula for permutations is<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image021.gif\" width=\"89\" height=\"44\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">This formula describes the\u00a0number of ways to arrange <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">elements out of <\/span><em><span style=\"text-decoration: none;\">n\u00a0<\/span><\/em><span style=\"text-decoration: none;\">elements. <\/span><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>There are 36 people in the 6th grade class. How many different\u00a0ways can the teacher line up 14 students?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image024.gif\" width=\"21\" height=\"34\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image027.gif\" width=\"105\" height=\"11\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image030.gif\" width=\"21\" 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\/s2_p2_clip_image033.gif\" width=\"106\" height=\"11\" name=\"graphics13\" 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 D. Instead of using the counting\u00a0principle directly and writing 14 blanks in which to label\u00a0possible choices, we can save some paper and employ the\u00a0permutation formula. Because order matters, we need to use the permutation formula.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image036.gif\" width=\"377\" height=\"42\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p><span style=\"text-decoration: none;\">Notice that if we plan to arrange <\/span><em><span style=\"text-decoration: none;\">all<\/span><\/em><span style=\"text-decoration: none;\"> the elements of our collection, we are back to the\u00a0original statement of the fundamental counting principle. <\/span><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image039.gif\" width=\"146\" height=\"42\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>, since 0! = 1.<\/p>\n<p class=\"notebox_text\" align=\"center\">In our example of the 7 books on a shelf, we initially arranged all 7\u00a0books.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p2_clip_image042.gif\" width=\"268\" height=\"42\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<h3><strong>How important is the issue of ordering?<\/strong><\/h3>\n<p>Order is very important. For example, you can have all of the\u00a0numbers correct for a safe combination, but you will not be able\u00a0to open it unless you have them in the correct order. As we saw\u00a0earlier, permutations count the elements of a sample space in\u00a0which the ordering matters, such as students in a line, books on a\u00a0shelf, or the order in which people finish a race. To count\u00a0elements in a sample space in which the order does not matter, we\u00a0use combinations.<\/p>\n<p>The formula for combinations is<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p3_clip_image003.gif\" width=\"100\" height=\"42\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>This formula describes the number of ways t<span style=\"text-decoration: none;\">o\u00a0choose <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">elements out of <\/span><em><span style=\"text-decoration: none;\">n\u00a0<\/span><\/em><span style=\"text-decoration: none;\">elements. <\/span><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Two restaurants in town offer vegetarian plates and a variety\u00a0of vegetables from which to choose. Restaurant A offers 6\u00a0different vegetables and customers can choose 2 per plate;\u00a0restaurant B offers 5 vegetables and lets customers choose 3 per\u00a0plate. Which statement below is correct?<\/p>\n<ol>\n<li>5 more vegetarian plates are possible at restaurant B than are\u00a0possible at restaurant A.<\/li>\n<li>3 more vegetarian plates are possible at restaurant A than are\u00a0possible at restaurant B.<\/li>\n<li>3 more vegetarian plates are possible at restaurant B than are\u00a0possible at restaurant A.<\/li>\n<li>5 more vegetarian plates are possible at restaurant A than are\u00a0possible at restaurant B.<\/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. The order doesn\u2019t matter when\u00a0serving vegetables on a plate, so we should use the combination\u00a0formula. The number of possible vegetarian plates at restaurant A\u00a0is:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p3_clip_image006.gif\" width=\"206\" height=\"49\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Similarly, the number of possible vegetarian plates at\u00a0restaurant B is:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p3_clip_image009.gif\" width=\"203\" height=\"49\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3>How can I tell when to use permutations and when to use<br \/>\ncombinations?<\/h3>\n<p>Because\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p4_clip_image003.gif\" width=\"62\" height=\"16\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0we must be able to determine which operation to use when counting\u00a0the total elements of a sample space. This choice hinges on the\u00a0question of ordering.<\/p>\n<p>Consider a desired result. If changing the order does not\u00a0change your desired result, then use combinations. If your desired\u00a0result depends on the order, use permutations.<\/p>\n<p><span style=\"text-decoration: none;\">So if 10 people are running\u00a0a race and we want to know which <\/span><em><span style=\"text-decoration: none;\">group\u00a0<\/span><\/em><span style=\"text-decoration: none;\">of 3 is the first across the finish line, we would use\u00a0combinations. If we want to know exactly who came in 1st, 2nd, and\u00a03rd, we would use permutation<\/span>s.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>Make a note of the following, it\u2019s worth memorizing!<\/p>\n<p class=\"notebox_text\" align=\"LEFT\"><span style=\"text-decoration: none;\">When order <\/span><em><span style=\"text-decoration: none;\">does<\/span><\/em><span style=\"text-decoration: none;\"> matter, use permutations <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p4_clip_image006.gif\" width=\"89\" height=\"44\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>. Think &#8220;<\/span><em><span style=\"text-decoration: none;\">n<\/span><\/em><span style=\"text-decoration: none;\"> arranging <\/span><em><span style=\"text-decoration: none;\">r<\/span><\/em><span style=\"text-decoration: none;\">.&#8221; <\/span><\/p>\n<p class=\"notebox_text\" align=\"LEFT\"><span style=\"text-decoration: none;\">For groups where order <\/span><em><span style=\"text-decoration: none;\">does not<\/span><\/em><span style=\"text-decoration: none;\"> matter, use combinations <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s2_p4_clip_image009.gif\" width=\"100\" height=\"42\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span>.<span style=\"text-decoration: none;\"> Think &#8220;<\/span><em><span style=\"text-decoration: none;\">n<\/span><\/em><span style=\"text-decoration: none;\"> choosing <\/span><em><span style=\"text-decoration: none;\">r<\/span><\/em><span style=\"text-decoration: none;\">.&#8221; <\/span><\/p>\n<\/div>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/defining-probability\">\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\/utilizing-combinations-and-permutations\">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 Combinations and Permutations Objective In this lesson, you will study how to use combinations and permutations. Previously Covered: The !\u00a0operation is called factorial. Combinations and Permutations Two other useful techniques that aid in counting are\u00a0permutations and combinations. In the earlier example, we wanted to find the ways that 7 [&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-295","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/295","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=295"}],"version-history":[{"count":6,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/295\/revisions"}],"predecessor-version":[{"id":628,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/295\/revisions\/628"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=295"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}