{"id":1806,"date":"2012-05-18T12:09:29","date_gmt":"2012-05-18T19:09:29","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=1806"},"modified":"2020-01-15T10:50:44","modified_gmt":"2020-01-15T18:50:44","slug":"gmat-quant-how-to-count","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-quant-how-to-count\/","title":{"rendered":"GMAT Quant: How to Count"},"content":{"rendered":"

\"student<\/p>\n

The title of this post may seem facetious.\u00a0After all, even the person most allergic to math, most traumatized by math, still remembers how to count!\u00a0 <\/p>\n

The GMAT, of course, generally will not ask you, for example, to count from one to seven.\u00a0The GMAT may give you a more complex scenario, and ask you to count how many ways can such-and-such happen.\u00a0 For example:<\/p>\n

Shakespeare wrote fifteen comedies (including the so-called “romances”), ten histories, and twelve tragedies. \u00a0If a summer Shakespeare festival always has one comedy, one history, and two tragedies, how many different combinations of plays can the festival host?<\/p><\/blockquote>\n

As you see, this “counting” is a little more challenging than the kind of “counting” you learned in your salad days.\u00a0 I would like to convince you, though, that you are quite capable of solving problems like this.<\/p>\n

 <\/p>\n

The Fundamental Counting Principle<\/h2>\n

This one big idea will give you a lot of mileage on any of the problems where the GMAT asks you to count things.<\/p>\n

If option #1 has P alternatives and option #2 has Q alternatives (assuming that the two sets of alternatives have no overlap), then total number of different pairs we can form is P*Q.\u00a0 For example: Shakespeare wrote fifteen comedies and ten histories.\u00a0 If we want to select one comedy and one history, the total number of possible pairs is 15*10 = 150.<\/p>\n

The FCP easily extends from two choices to three or any higher number.\u00a0 However many collections of alternatives there are, you simply multiple the number of alternatives in each set to produce the total number of combinations.\u00a0 For example: Shakespeare wrote fifteen comedies, ten histories, and twelve tragedies.\u00a0 If we are going to pick one of each kind, and ask how many different trios of plays can we create, the total number is simply 15*10*12.\u00a0 BTW, figuring that out without a calculator is not so hard.\u00a0 First, on the 15*12, use the “doubling & halving<\/a>” trick —– 15 times 2 is 30, and 12 divided by 2 is 6, so 15*12 = 30*6 = 180, and last, the easiest of all, multiplying by ten: 15*12*10 = 1800.<\/p>\n

 <\/p>\n

Permutations and Combinations<\/h2>\n

Before we can answer the original question posed, we have to clarify some terminology about counting.\u00a0 A permutation<\/strong> is a set in which order matters —- AABC is a different permutation from BACA.\u00a0 A combination<\/strong> is a set in which order does not matter —- AABC and BACA are the same combination of four letters.<\/p>\n

This distinction is important in counting because we have to know whether to include the sets that repeat elements in different order.\u00a0 In question #1 above, the question explicitly asks for combinations.\u00a0 In other words, if we pick Hamlet<\/em> and then King Lear<\/em>, that’s will be considered the same as picking King Lear<\/em> and then Hamlet<\/em>.<\/p>\n

This post:\u00a0https:\/\/magoosh.com\/gmat\/math\/gmat-permutations-and-combinations\/<\/a>\u00a0considers permutations and combinations in greater detail.\u00a0 For the purposes of this post, we just have to be careful to consider what we are counting.<\/p>\n

 <\/p>\n

How Many Pairs of Tragedies?<\/h2>\n

Shakespeare wrote twelve tragedies, and we want to pick a pair from these.\u00a0 How many different pairs can we pick?\u00a0 We will use the FCP.<\/p>\n

Let\u2019s break the task into a first choice and a second choice.\u00a0 For the first choice, we can choose any of the 12 tragedies, so we have 12 choices.\u00a0 Suppose, for the sake of argument, we pick Romeo and Juliet<\/em> on the first choice.\u00a0 Now, for the second choice, notice that we don’t still have 12 choices, because we don’t want to pick Romeo and Juliet<\/em> again on the second choice.\u00a0 In general, on the second choice, you have one fewer choice than you had on the first choice, because you don’t want to duplicate whatever element was chosen the first time.\u00a0 Thus, on the second choice there are eleven choices.\u00a0 By the FCP, it would seem the total number of possible pairs of would be 12*11.<\/p>\n

If we were interested in permutations, then 12*11 would be the correct answer.\u00a0 Here, though, we are interested in combinations, not permutations.\u00a0 Again, if we pick Romeo and Juliet<\/em>, then Macbeth<\/em>, we will count that as the same pair as Macbeth<\/em> first, followed by Romeo and Juliet<\/em>.\u00a0 The figure of 12*11 automatically counts each permutations, and so counts every pair twice, as AB and then as BA.\u00a0 Thus, we have to divide 12*11 by two.\u00a0 Thus, 12*11\/2 = 6*11 = 66 is the number of possible pairs of tragedies.<\/p>\n

 <\/p>\n

The Whole Shebang<\/h2>\n

Now, we can answer the entire question.\u00a0 We want the number of combinations of four plays consisting of one comedy, one history, and two tragedies.\u00a0 By the FCP, that’s 15*10*66.\u00a0 Notice the math tricks to multiply easily without a calculator.<\/p>\n

Split 66 back into 6*11 —- 15*10*6*11<\/p>\n

Switch the order —– 15*6*11*10<\/p>\n

Use doubling-halving<\/a> on the 15 & 6 —- (15*6)*11*10 =\u00a0 (30*3)*11*10 = 90*11*10<\/p>\n

Then 90*11 —- 90*11*10 = 990*10<\/p>\n

Then, the easiest, multiplying by ten — 990*10 = 9900<\/p>\n

Thus, the festival can come up with 9900 combinations consisting of one comedy, one history, and two tragedies.<\/p>\n

 <\/p>\n

Practice Question<\/h2>\n

Here’s a practice question on this very topic:\u00a0https:\/\/gmat.magoosh.com\/questions\/845<\/a><\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

The title of this post may seem facetious.\u00a0After all, even the person most allergic to math, most traumatized by math, still remembers how to count!\u00a0 The GMAT, of course, generally will not ask you, for example, to count from one to seven.\u00a0The GMAT may give you a more complex scenario, and ask you to count […]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[150],"tags":[],"ppma_author":[13209],"class_list":["post-1806","post","type-post","status-publish","format-standard","hentry","category-basics"],"acf":[],"yoast_head":"\nGMAT Quant: How to Count - 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-quant-how-to-count\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"GMAT Quant: How to Count\" \/>\n<meta property=\"og:description\" content=\"The title of this post may seem facetious.\u00a0After all, even the person most allergic to math, most traumatized by math, still remembers how to count!\u00a0 The GMAT, of course, generally will not ask you, for example, to count from one to seven.\u00a0The GMAT may give you a more complex scenario, and ask you to count […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/gmat\/gmat-quant-how-to-count\/\" \/>\n<meta property=\"og:site_name\" content=\"Magoosh Blog \u2014 GMAT\u00ae Exam\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/MagooshGMAT\/\" \/>\n<meta property=\"article:published_time\" content=\"2012-05-18T19:09:29+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-01-15T18:50:44+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/magoosh.com\/gmat\/files\/2012\/05\/10.png\" \/>\n<meta name=\"author\" content=\"Mike M\u1d9cGarry\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:site\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Mike M\u1d9cGarry\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-how-to-count\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-how-to-count\/\"},\"author\":{\"name\":\"Mike M\u1d9cGarry\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/320346c205075513344435baf9b0521b\"},\"headline\":\"GMAT Quant: How to Count\",\"datePublished\":\"2012-05-18T19:09:29+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-how-to-count\/\"},\"wordCount\":833,\"commentCount\":9,\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/#organization\"},\"articleSection\":[\"GMAT Math Basics\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-how-to-count\/\",\"url\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-how-to-count\/\",\"name\":\"GMAT Quant: How to Count - 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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\/1806","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=1806"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/1806\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=1806"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=1806"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=1806"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=1806"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}