{"id":14414,"date":"2014-01-20T09:00:01","date_gmt":"2014-01-20T17:00:01","guid":{"rendered":"https:\/\/magoosh.com\/gre\/?p=14414"},"modified":"2019-07-04T12:23:57","modified_gmt":"2019-07-04T19:23:57","slug":"does-order-matter-combinations-vs-the-fundamental-counting-principle-on-the-gre","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gre\/does-order-matter-combinations-vs-the-fundamental-counting-principle-on-the-gre\/","title":{"rendered":"Does Order Matter? Combinations vs. the Fundamental Counting Principle on the GRE"},"content":{"rendered":"<p>When trying to decide whether to use the combination formula, many students have trouble applying the &#8220;does order matter&#8221; test&#8211;deciding if outcomes differ. It\u2019s easy to say order doesn\u2019t matter when it in fact does, and vice versa. Once you get it, it&#8217;s really useful, but it can take a bit of thinking and re-thinking.<\/p>\n<p>Let\u2019s look at a couple of examples to illustrate.<\/p>\n<p>&nbsp;<\/p>\n<h2>A Simple Case of the FCP<\/h2>\n<p>Suppose we have 5 people waiting for 3 seats. Now let&#8217;s say I want to count how many ways 3 people can be arranged in those seats. There are three tasks here&#8211;one for each seat.<\/p>\n<p><i>fill 1st seat, fill 2nd seat, fill 3rd seat<\/i><\/p>\n<p>We can write those as 3 blanks:<\/p>\n<p>_____ , _____ , _____<\/p>\n<p>With this done, we&#8217;re most of the way there. We just need to check if filling one seat with one person is the same as filling another seat with that same person.<\/p>\n<p><span style=\"text-decoration: underline\">Amy<\/span> , _____ , _____<\/p>\n<p>_____ , <span style=\"text-decoration: underline\">Amy<\/span> , _____<\/p>\n<p>Now, we know those aren&#8217;t the same, because that makes two different line-ups. Keep in mind which task each blank represents.<b> <\/b><i>Amy in seat 1 is not the same as Amy in seat 2<\/i><b><i>.<\/i><\/b> So this is <i>not<\/i> a combination. Instead, we can use the fundamental counting principle (FCP).<\/p>\n<p><span style=\"text-decoration: underline\">\u00a0 5 options\u00a0 <\/span>, <span style=\"text-decoration: underline\">\u00a0\u00a04 options\u00a0 <\/span>, <span style=\"text-decoration: underline\">\u00a0\u00a03 options\u00a0 <\/span>\u00a0\u00a0<span style=\"text-decoration: underline\">\u00a0<\/span><\/p>\n<p>5*4*3 = 60.<\/p>\n<p>If we were simply looking for a group of people&#8211;not paying attention to which person takes which chair, then we&#8217;d use a combination because the tasks aren&#8217;t inherently separate. There would be no significance of which blank represents which task. That might be asking &#8220;how many ways three people can be chosen for a committee from a group of five,&#8221; for example. In that case, we <i>can&#8217;t<\/i> draw the tasks as separate. Selecting the first person for the committee is the same as as selecting the second person. We could even choose all three at the same time, in one task&#8211;we don&#8217;t have to break it up into separate events. In that case, we have to use the combination formula as given in the our combination <a href=\"http:\/\/gre.magoosh.com\/lessons\">lesson videos<\/a>.<\/p>\n<p>5C3<\/p>\n<p>5! \/ ([5-3]! *3!)<\/p>\n<p>5! \/ ([2]! *3!)<\/p>\n<p>5*4 \/ 2<\/p>\n<p>10<\/p>\n<p>&nbsp;<\/p>\n<h2>A Permutation that <i>Looks<\/i> like a Combination<\/h2>\n<p>Here&#8217;s another example. If you have 4 shirts, 4 pairs of pants, and 4 hats, and you choose one of each, is it a combination or does order matter? It&#8217;s tempting to say &#8220;order doesn&#8217;t matter&#8221;, because a t-shirt and jeans is the same as jeans and a t-shirt, but let&#8217;s try making those 3 tasks:<\/p>\n<p><i>pick a shirt, pick pants, pick a hat<\/i><\/p>\n<p>or<\/p>\n<p>_____ , _____ , _____<\/p>\n<p>Again, remember that each blank must represent one task alone, and we don&#8217;t move the blanks around. So if we say<\/p>\n<p><span style=\"text-decoration: underline\">red t-shirt<\/span>, _____ , _____<\/p>\n<p>That&#8217;s NOT the same as<\/p>\n<p>_____ , _____ , <span style=\"text-decoration: underline\">red t-shirt<\/span><\/p>\n<p><i>A shirt in the first blank is not the same as a shirt in the last blank,<\/i> because the last blank is for choosing a <i>hat<\/i> and a shirt cannot be a hat. It&#8217;s not very fashionable at the moment, at least.<\/p>\n<p>That means that we can use the fundamental counting principle here. First, we look at the shirt task. There are 4 shirts, so we have 4 possibilities. Next, and <i>separately, <\/i>we look at the pants. There are 4 pairs, so that&#8217;s another 4 possibilities for each shirt. So far, that&#8217;s 4*4. And finally, we have hats: there are another 4 hats, so that&#8217;s another 4 possibilities for each match up of shirt and pants. That&#8217;s 4*4*4.<\/p>\n<p>So if we have <i>separate<\/i> blanks for each task, just fill in the blank with how many choices can be made in that specific step. Then multiply all the numbers for your answer.<\/p>\n<p>On the other hand, if we were looking for ANY three pieces of clothing from the twelve total, we would use a combination formula. Again, this could be done in a single step&#8211;reach into a bag of clothes and grab three things. There&#8217;s no differentiation between them. This would give us 12C3, or 220.<\/p>\n<p>Basically, this is all about drawing the blanks to represent the tasks. As you do so, ask if the item or person you\u2019re picking can be moved around to different blanks without changing the situation. (Move the items&#8211;not the blanks!). If the item can be moved without changing the situation, it&#8217;s a combination&#8211;order doesn&#8217;t matter. If moving the item changes the situation (like in the line-up) or is impossible (like with the clothes), then we use the FCP.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>When trying to decide whether to use the combination formula, many students have trouble applying the &#8220;does order matter&#8221; test&#8211;deciding if outcomes differ. It\u2019s easy to say order doesn\u2019t matter when it in fact does, and vice versa. Once you get it, it&#8217;s really useful, but it can take a bit of thinking and re-thinking. [&hellip;]<\/p>\n","protected":false},"author":42,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[39,25],"tags":[],"ppma_author":[12270],"class_list":["post-14414","post","type-post","status-publish","format-standard","hentry","category-data-analysis","category-math"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Does Order Matter? Combinations vs. the Fundamental Counting Principle on the GRE - Magoosh Blog \u2014 GRE\u00ae Test<\/title>\n<meta name=\"description\" content=\"Having trouble applying the \u201cdoes order matter\u201d test when deciding to use the combination formula? 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