{"id":12881,"date":"2020-05-27T15:04:55","date_gmt":"2020-05-27T15:04:55","guid":{"rendered":"https:\/\/www.manhattanprep.com\/gre\/?p=12881"},"modified":"2020-05-27T15:06:21","modified_gmt":"2020-05-27T15:06:21","slug":"combinatorics-problems-on-the-gre","status":"publish","type":"post","link":"https:\/\/www.manhattanprep.com\/gre\/blog\/combinatorics-problems-on-the-gre\/","title":{"rendered":"How to Actually Do Combinatorics Problems on the GRE"},"content":{"rendered":"

\"Combinatorics<\/span><\/p>\n

Combinatorics\u2014it\u2019s a word none of us can say and none of us had ever heard of before we started studying for the GRE. It\u2019s a fancy word that just means \u201cthe number of possibilities\u201d or \u201call the ways something could go\u201d (my definitions).\u00a0<\/span><\/p>\n

<\/p>\n

And once we do start to tackle it, we\u2019re often stumped. Most books and online study tools teach us a formula that is deceptively simple. \u201cGreat!\u201d we think. \u201cI\u2019m set!\u201d\u00a0<\/span><\/p>\n

Then we get to a combinatorics problem on a practice test, and what happens? We get it wrong. The formula doesn\u2019t work.\u00a0<\/span><\/p>\n

It\u2019s a frustrating experience, and one I\u2019m going to solve for you in this post.\u00a0<\/span><\/p>\n

First, though, a caveat\u2014there are very few of these problems on the test. It\u2019s not worth your time to spend hours studying combinatorics over, say, algebra, or percentage word problems. So while I want you to be empowered to answer combinatorics questions, I also don\u2019t want you to overinvest your time at the expense of working on other areas that you\u2019re likely to be tested on…like geometry.\u00a0<\/span><\/p>\n

GRE Combinatorics Strategies<\/b><\/h3>\n

Got it? Okay. Now let\u2019s cover the combinatorics strategy that we teach at Manhattan Prep because it\u2019s actually usable on GRE combinatorics problems. It\u2019s not a simple formula, but unlike the simple formula, it\u2019s applicable on the test.\u00a0<\/span><\/p>\n

First, two key terms: slots and labels.<\/span><\/p>\n

Slots – <\/b>a blank line for every decision\u00a0<\/span><\/p>\n

Labels – <\/b>description of the category of item that will fill the slot<\/span><\/p>\n

Here\u2019s your process:<\/span><\/p>\n

Step 1: <\/b>Create slots for every decision you\u2019re making.\u00a0<\/span><\/p>\n

Step 2: <\/b>Label those slots according to the type of thing that will fill them.\u00a0\u00a0<\/span><\/p>\n

Step 3:<\/b> Enter the number of options for each slot.\u00a0<\/span><\/p>\n

Step 4:<\/b> Multiply to find the total number of possibilities.\u00a0<\/span><\/p>\n

That\u2019s (almost) it! We are 90% of the way done.<\/span><\/p>\n

Let\u2019s take a look at an example.<\/span><\/p>\n

Suppose we are creating a singing group of four people, choosing from a group of six singers. Each person will have an assigned role: soprano, alto, tenor, and bass. We make four slots for each of the parts, and label them:\u00a0<\/span><\/p>\n

\"GRE<\/p>\n

Now, we fill the slots with the number of options for each decision. For the first spot, we are choosing from 6 people: <\/span><\/p>\n

\"GRE<\/p>\n

For the second, we\u2019re now choosing from <\/span>5<\/span><\/i> people\u2014one person has already been chosen:<\/span><\/p>\n

\"GRE<\/p>\n

Let\u2019s finish filling them in:<\/span><\/p>\n

\"GRE<\/p>\n

From here, we are almost done but not quite. We must consult our labels and ask: <\/span>Are all of the labels different, or do any repeat? <\/b>If they\u2019re all different\u2014which they are in this case\u2014we\u2019re done! We simply multiply to find the total number of possible book groups:\u00a0<\/span><\/p>\n

6 x 5 x 4 x 3 =\u00a0 360 possible combinations<\/strong><\/p>\n

But what if we have repeated labels? In this case, we add one more step:<\/span><\/p>\n

Step 1<\/b>: Create slots for every decision you\u2019re making.\u00a0<\/span><\/p>\n

Step 2: <\/b>Label those slots according to the type of thing that will fill them.\u00a0\u00a0<\/span><\/p>\n

Step 3:<\/b> Enter the number of options for each slot.\u00a0<\/span><\/p>\n

Step 4: <\/b>Multiply to find the total number of possibilities.\u00a0<\/span><\/p>\n

Step 5: <\/b>Ask\u2014do any of the labels repeat? If no, you\u2019re done. If yes…<\/span><\/p>\n

Say, for example, that we\u2019re just making a singing group with <\/span>no <\/span><\/i>assigned roles? So the labels aren\u2019t Soprano, Alto, Tenor, and Bass, they\u2019re just Singer (I\u2019ll use \u201cS\u201d):<\/span><\/p>\n

\"GRE<\/p>\n

Now, we have the same label\u2014S\u2014used more than once. <\/span>Whenever you have a label used more than once, you must take one last step: divide by the factorial of the number of repeated labels. <\/b>(The reason for this is that we have overcounted\u2014360 is going to be too big. If you want a challenge, see if you can figure out why!)\u00a0<\/span><\/p>\n

In this case, we have <\/span>four <\/span><\/i>repeated Ss, so we will divide by 4 factorial:<\/span><\/p>\n

\"combinatorics<\/p>\n

And that\u2019s our method!<\/span><\/p>\n

GRE Combinatorics Strategy Review<\/b><\/h3>\n

Let\u2019s review:\u00a0<\/span><\/p>\n

Step 1: <\/b>Create slots for every decision you\u2019re making.\u00a0<\/span><\/p>\n

Step 2: <\/b>Label those slots according to the type of thing that will fill them.\u00a0\u00a0<\/span><\/p>\n

Step 3: <\/b>Enter the number of options for each slot.\u00a0<\/span><\/p>\n

Step 4: <\/b>Multiply to find the total number of possibilities.\u00a0<\/span><\/p>\n

Step 5:<\/b> Ask\u2014do any of the labels repeat? If no, you\u2019re done. If yes, divide by the factorial of the number of repeated labels.<\/span><\/p>\n

One last thing\u2014if you have multiple repeated labels, let\u2019s say 4 S\u2019s and and 3 Y\u2019s (just randomly picking letters here), just divide by the factorial of both: 4! x 3!, or: 4 x 3 x 2 x 2 x 3 x 2 x 1.\u00a0<\/span><\/p>\n

Happy combinatorics-ing!\u00a0<\/span><\/p>\n

READ NEXT: <\/strong>Why a Review Log is Vital to Your GRE Prep<\/a><\/p>\n

Don\u2019t forget that you can attend the first session of any of our online GRE courses absolutely free. We\u2019re not kidding!\u00a0<\/i><\/b>Check out our upcoming courses here<\/i><\/b><\/a>.<\/i><\/b><\/p>\n

\n

\"Mary<\/p>\n

Mary Richter is a Manhattan Prep instructor based in Nashville, Tennessee.\u00a0<\/i><\/b>Mary is one of those weirdos who loves taking standardized tests, and she has been teaching them for 15 years. When she\u2019s not teaching the LSAT or GRE for ManhattanPrep, she\u2019s writing novels under the last name Adkins. You can find them wherever you buy books.\u00a0<\/i>Check out Mary\u2019s upcoming GRE prep offerings here!\u00a0<\/i><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Combinatorics\u2014it\u2019s a word none of us can say and none of us had ever heard of before we started studying for the GRE. It\u2019s a fancy word that just means \u201cthe number of possibilities\u201d or \u201call the ways something could go\u201d (my definitions).\u00a0<\/p>\n","protected":false},"author":183,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[1365390,133,1365388,151,161,1365389],"yst_prominent_words":[1365377,1365370,1365379,1365387,1365380,1365374,1365376,1365365,1365378,1365364,1365375,1365383,1365382,1365373,1365369,1365371,1365372,1365381,1365386,1365385],"class_list":["post-12881","post","type-post","status-publish","format-standard","hentry","category-gre-strategies","tag-combinatorics","tag-gre","tag-gre-combinatorics","tag-gre-math","tag-gre-quant","tag-gre-quantitative"],"_links":{"self":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts\/12881","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/users\/183"}],"replies":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/comments?post=12881"}],"version-history":[{"count":4,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts\/12881\/revisions"}],"predecessor-version":[{"id":12893,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts\/12881\/revisions\/12893"}],"wp:attachment":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/media?parent=12881"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/categories?post=12881"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/tags?post=12881"},{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/yst_prominent_words?post=12881"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}