{"id":2893,"date":"2012-10-18T09:00:26","date_gmt":"2012-10-18T16:00:26","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=2893"},"modified":"2020-01-15T10:50:23","modified_gmt":"2020-01-15T18:50:23","slug":"gmat-factorials","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-factorials\/","title":{"rendered":"GMAT Factorials"},"content":{"rendered":"<p><strong>Learn how to simplify these seemingly devilishly complicated GMAT Quant problems!<\/strong><\/p>\n<p>First, consider these problems<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img1.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2895\" title=\"sf_img1\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img1.png\" alt=\"\" width=\"198\" height=\"145\" \/><\/a><\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img2.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2896\" title=\"sf_img2\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img2.png\" alt=\"\" width=\"138\" height=\"145\" \/><\/a><\/p>\n<p>3) Consider these three quantities<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img3.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2897\" title=\"sf_img3\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img3.png\" alt=\"\" width=\"155\" height=\"172\" \/><\/a><\/p>\n<p>Rank these three quantities from least to greatest.<\/p>\n<ul>\n<ul>(A) I, II, III<\/ul>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<ul>(B) I, III, II<\/ul>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<ul>(C) II, I, III<\/ul>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<ul>(D) II, III, I<\/ul>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>(E) III, I, II<\/ul>\n<p>These are challenging problems, especially the third one!\u00a0 With a few simple insights about factorials, though, you will be able to manage all of these.<\/p>\n<p>&nbsp;<\/p>\n<h2>The factorial<\/h2>\n<p>First of all, a few basics.\u00a0 The factorial is a function we can perform on any positive integer.\u00a0 The expression 5! (read &#8220;five factorial&#8221;) means the product of all the positive integers from that number down to one.\u00a0 In this particular case:\u00a0 5! = 5*4*3*2*1 = 120.\u00a0 Here are the first ten factorials, just to give you a sense:<\/p>\n<p>1! = 1<\/p>\n<p>2! = 2*1 = 2<\/p>\n<p>3! = 3*2*1 = 6<\/p>\n<p>4! = 4*3*2*1 = 24<\/p>\n<p>5! = 5*4*3*2*1 = 120<\/p>\n<p>6! = 6*5*4*3*2*1 = 720<\/p>\n<p>7! = 7*6*5*4*3*2*1 = 5,040<\/p>\n<p>8! = 8*7*6*5*4*3*2*1 = 40,320<\/p>\n<p>9! = 9*8*7*6*5*4*3*2*1 = 362,880<\/p>\n<p>10! = 10*9*8*7*6*5*4*3*2*1 = 3,628,800<\/p>\n<p>It&#8217;s a good idea to have the first five memorized, simply because those come up frequently on the GMAT &#8212;the first five are pretty easy to figure out on your own anyway.\u00a0 <em><span style=\"text-decoration: underline\">Nobody<\/span><\/em> expects you to have the last five here memorized.\u00a0 I give them here purely to give you a sense of how quickly the factorials grow.\u00a0 By the time we get to (10!), we are already over a million!\u00a0 Similarly, (13!) is more than a billion, and (15!) is more than a trillion.\u00a0 Holy schnikes!<\/p>\n<p>&nbsp;<\/p>\n<h2>Operations with factorials<\/h2>\n<p>Here are some big ideas to help you when you have to perform arithmetic operations with factorials.<\/p>\n<p><strong>Big Idea #1: every factorial is a factor of every higher factorial<\/strong><\/p>\n<p>The number 73! must be divisible by 72!, by 47!, by 12!, etc.\u00a0\u00a0 The number 73! automatically has at least 72 known factors &#8212; all the factorials less than it!<\/p>\n<p><strong>Big Idea #2: you can &#8220;unpack&#8221; one factorial down to another.\u00a0 <\/strong><\/p>\n<p>Think about 8! &#8212;- we know:<\/p>\n<p>8! = 8*7*6*5*4*3*2*1<\/p>\n<p>Well, by the Associative Law, we can group factors in any groupings, so we could insert parenthesis wherever we like.\u00a0 In particular , if I include\u00a0 a selection of factors that goes all the way to the right, all the way to 1, that&#8217;s another factorial.\u00a0 Thus:<\/p>\n<p>8! = 8*(7*6*5*4*3*2*1) = 8*(7!)<\/p>\n<p>8! = 8*7*(6*5*4*3*2*1) = 8*7*(6!)<\/p>\n<p>8! = 8*7*6*(5*4*3*2*1) = 8*7*6*(5!)\u00a0 etc.<\/p>\n<p>I chose 8! because we can see all the factors, but clearly we could extend this idea to any factorial, no matter how large:<\/p>\n<p>237! = 237*236*(235!)<\/p>\n<p>That&#8217;s an &#8220;unpacking&#8221; of the first two factors of 237!\u00a0\u00a0 (BTW, this term, &#8220;unpacking&#8221;, is my own creation: you will not see this term used anywhere else.)<\/p>\n<p><strong>Big Idea #3: when you <span style=\"text-decoration: underline\">divide<\/span> two factorials, you &#8220;unpack&#8221; the larger one, and cancel it with the smaller one. <\/strong><\/p>\n<p>Example:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img4.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2898\" title=\"sf_img4\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img4.png\" alt=\"\" width=\"356\" height=\"49\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img4.png 356w, https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img4-300x41.png 300w\" sizes=\"(max-width: 356px) 100vw, 356px\" \/><\/a><\/p>\n<p><strong>Big Idea #4: when you <span style=\"text-decoration: underline\">add\/subtract<\/span> two or more factorials, you unpack them all down to the lowest one, and factor out that common factor.\u00a0 <\/strong><\/p>\n<p>For more on &#8220;factoring out&#8221;, see the section &#8220;Distributing and factor out&#8221; in this <a href=\"https:\/\/magoosh.com\/gmat\/adding-and-subtracting-powers-on-the-gmat\/\">post<\/a>.<\/p>\n<p>Examples:<\/p>\n<p>(200!) \u2013 (199!) = 200*(199!) \u2013 (1)*(199!) = (200 \u2013 1)*(199!) = 199*(199!)<\/p>\n<p>(200!) + (199!) = 200*(199!) + (1)*(199!) = (200 + 1)*(199!) = 201*(199!)<\/p>\n<p>&nbsp;<\/p>\n<h2>Summary<\/h2>\n<p>Having read these rules, give those three practice problems another try before reading the solutions below.\u00a0 Here&#8217;s a fourth problem, with its own video explanation.<\/p>\n<p>4) <a href=\"http:\/\/gmat.magoosh.com\/questions\/811\">http:\/\/gmat.magoosh.com\/questions\/811<\/a><\/p>\n<p>&nbsp;<\/p>\n<h2>Practice problem solutions<\/h2>\n<p>1) We are going to use the &#8220;factoring out&#8221; trick in both the numerator and the denominator:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img5.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2899\" title=\"sf_img5\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img5.png\" alt=\"\" width=\"534\" height=\"52\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img5.png 534w, https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img5-300x29.png 300w\" sizes=\"(max-width: 534px) 100vw, 534px\" \/><\/a><\/p>\n<p>Now, &#8220;unpack&#8221; that top factorial, to cancel the smaller one in the denominator:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img6.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2900\" title=\"sf_img6\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img6.png\" alt=\"\" width=\"510\" height=\"54\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img6.png 510w, https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img6-300x31.png 300w\" sizes=\"(max-width: 510px) 100vw, 510px\" \/><\/a><\/p>\n<p>Answer = <strong>D<\/strong><\/p>\n<p>2) &#8220;Unpack&#8221; the factorials in the numerator, so that everything is expressed as a product involving (89!)\u00a0 Then factor out that common factor.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img7.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2901\" title=\"sf_img7\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img7.png\" alt=\"\" width=\"557\" height=\"90\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img7.png 557w, https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img7-300x48.png 300w\" sizes=\"(max-width: 557px) 100vw, 557px\" \/><\/a><\/p>\n<p>Answer = <strong>D<\/strong><\/p>\n<p>3) Let&#8217;s consider the three expressions separately.<\/p>\n<p>For expression I, we merely have to &#8220;unpack&#8221; the (49!) factor in the numerator.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img8.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2902\" title=\"sf_img8\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img8.png\" alt=\"\" \/><\/a><\/p>\n<p>So, expression I has a value of 49.<\/p>\n<p>The expression II is tricky:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img9.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2903\" title=\"sf_img9\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img9.png\" alt=\"\" \/><\/a><\/p>\n<p>That&#8217;s a whole lot of factors in the numerator!\u00a0 That numerator is fantastically big: it has forty factors from 49 to 10 that are all greater than or equal to 10, so that means it&#8217;s automatically bigger than 10^40 (a number <strong><em>bigger than 1 trillion cubed<\/em><\/strong> &#8212; <strong>OMG!<\/strong>) That&#8217;s divided by 7! = 5040, so whatever this is, it&#8217;s way bigger than 49.<\/p>\n<p>The expression III is a little easier than II:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img10.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2904\" title=\"sf_img10\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img10.png\" alt=\"\" \/><\/a><\/p>\n<p>That&#8217;s also a very big number.\u00a0 Each of the seven factors in the numerator is greater than 10, and 10^7 is ten million, so that numerator is more than 10,000,000.\u00a0 That&#8217;s divided by 7! = 5040.\u00a0 This is clearly bigger than 49.\u00a0 Notice, though, this has the same denominator as II, but many fewer factors in the numerator.\u00a0 Therefore II is much bigger than III.<\/p>\n<p>Thus, from least\u00a0to greatest, the order is I, III, II.\u00a0 Answer = <strong>B<\/strong><\/p>\n<p>Just as a note, if you are familiar with the idea of combinations, you may recognize expression II as a <a href=\"https:\/\/magoosh.com\/gmat\/gmat-permutations-and-combinations\/\">combinations<\/a> number:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img11.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2905\" title=\"sf_img11\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img11.png\" alt=\"\" \/><\/a><\/p>\n<p>That would be the number of unique sets of 7 we could select from a pool of 49 unique items.\u00a0 There is no way you would be expected to calculate that number without a serious calculator or computer.<\/p>\n<p>That&#8217;s a reasonably big number \u2013 just over 85 million. \u00a0That&#8217;s a little more than a quarter the current population of the USA.\u00a0 For comparison, expression II is a real whopper:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img12.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2906\" title=\"sf_img12\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/10\/sf_img12.png\" alt=\"\" \/><\/a><\/p>\n<p>This is a larger number than all humans and all other living things (animals &amp; plants &amp; all the way down to single-cell critters), including those alive now as well as those who have ever been alive on Earth.\u00a0 This number is larger than all the money in the world in pennies.\u00a0 This is more than the number of individual atoms comprising planet Earth and everything on Earth.\u00a0 This number is much much larger than the number of stars &amp; planets &amp; pulsars &amp; quasars &amp; black holes &amp; whatever other star-like things in all galaxies &amp; clusters in the visible Universe.\u00a0 The phrase &#8220;inconceivably big&#8221; does not even begin to capture how big this number is!<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Learn how to simplify these seemingly devilishly complicated GMAT Quant problems! First, consider these problems 3) Consider these three quantities Rank these three quantities from least to greatest. (A) I, II, III &nbsp; (B) I, III, II &nbsp; (C) II, I, III &nbsp; (D) II, III, I &nbsp; (E) III, I, II These are challenging [&hellip;]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[160],"tags":[],"ppma_author":[13209],"class_list":["post-2893","post","type-post","status-publish","format-standard","hentry","category-arithmetic"],"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>GMAT Factorials - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gmat\/gmat-factorials\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"GMAT Factorials\" \/>\n<meta property=\"og:description\" content=\"Learn how to simplify these seemingly devilishly complicated GMAT Quant problems! 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First, consider these problems 3) Consider these three quantities Rank these three quantities from least to greatest. 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Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations.","sameAs":["https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured"],"award":["Magna cum laude from Harvard"],"knowsAbout":["GMAT"],"knowsLanguage":["English"],"jobTitle":"Content Creator","worksFor":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/author\/mikemcgarry\/"}]}},"authors":[{"term_id":13209,"user_id":26,"is_guest":0,"slug":"mikemcgarry","display_name":"Mike M\u1d9cGarry","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","user_url":"","last_name":"M\u1d9cGarry","first_name":"Mike","description":"Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's <a href=\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\" rel=\"noopener noreferrer\">Youtube <\/a>video explanations and resources like <a href=\"https:\/\/magoosh.com\/gmat\/whats-a-good-gmat-score\/\" rel=\"noopener noreferrer\">What is a Good GMAT Score?<\/a> and the <a href=\"https:\/\/magoosh.com\/gmat\/gmat-diagnostic-test\/\" rel=\"noopener noreferrer\">GMAT Diagnostic Test<\/a>."}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/2893","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=2893"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/2893\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=2893"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=2893"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=2893"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=2893"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}