{"id":15668,"date":"2018-05-03T18:58:01","date_gmt":"2018-05-03T18:58:01","guid":{"rendered":"https:\/\/www.manhattanprep.com\/gmat\/?p=15668"},"modified":"2019-08-30T17:35:58","modified_gmt":"2019-08-30T17:35:58","slug":"everything-need-know-combinatorics-gmat","status":"publish","type":"post","link":"https:\/\/www.manhattanprep.com\/gmat\/blog\/everything-need-know-combinatorics-gmat\/","title":{"rendered":"Everything You Need to Know about Combinatorics for the GMAT"},"content":{"rendered":"

\"Manhattan<\/p>\n

Guess what? You can attend the first session of any of our online or in-person GMAT courses absolutely free\u2014we\u2019re not kidding!\u00a0<\/i><\/b>Check out our upcoming courses here<\/i><\/b><\/a>.<\/i><\/b><\/p>\n

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It\u2019s a pretty common question we GMAT teachers get: \u201cCan we go over combinatorics?\u201d To which my answer is usually a barely contained sigh.<\/span><\/p>\n

Sure we can, but the reason you want to go over combinatorics is because they\u2019re hard, and you\u2019re under the mistaken impression that you need to get hard questions right to do well on the GMAT\u2014and this just isn\u2019t really that true. You get a 700 score more by not missing many 600-level questions than by getting 700-level questions correct. And the fact is, you\u2019re just not going to see many combinatorics questions on the test, at any level.<\/span><\/p>\n

\u201cBut if I see a 600-level combinatorics question, and you said I need to not miss those to get a 700, don\u2019t I\u2014\u201d<\/span><\/p>\n

Yeah, yeah, yeah, I heard myself. You still want to make sure you master the more common topics first: percents, algebraic translations, exponent rules, number properties, testing cases on Data Sufficiency, and the like. But if you\u2019re pretty good on that, sure, let\u2019s get the basics of combinatorics questions down so that you can get them right, as well.<\/span><\/p>\n

So that\u2019s what this post is meant to be: the definitive guide to getting combinatorics problems through the 600 level. By the end, you\u2019ll have some of the most essential formulas for these questions, but also, I hope, a deeper understanding of the relationships <\/span>between<\/span><\/i> these formulas.<\/span><\/p>\n

The Basics on GMAT Combinatorics<\/b><\/h4>\n

Let\u2019s start with the most basic rule involved in combinatorics problems. It\u2019s actually got a name: \u201cThe Fundamental Counting Principle.\u201d This rule states that in order to determine the total number of overall outcomes, you multiply the number of each discrete outcome together. Or put more simply, \u201cIf I have X options in one case, and Y options in another, I have X*Y total options.\u201d<\/span><\/p>\n

So if I have four choices of sandwich, ten choices of chips, and three choices of snake venom soda, I have 4*10*3=120 total options for my lunch.<\/span><\/p>\n

USE THIS WHEN: You\u2019re asked about total number of outcomes from different choices and the choices do not affect one another. <\/span><\/p>\n

Okay, Let\u2019s Ramp it Up a Little<\/b><\/h4>\n

The next thing you\u2019re going to want to know about is <\/span>permutations<\/span><\/i>.<\/span><\/p>\n

A permutation is, quite simply, an arrangement of items. Let\u2019s use a classic game to discuss this topic: Mario Kart 64.<\/span><\/p>\n

In Mario Kart 64, every race had 8 racers. How many outcomes were possible for a race? <\/span><\/p>\n

This is a classic arrangement problem. Things are put in a certain order, and every order is considered different. Well, we can solve this by using the Fundamental Counting Principle. With 8 racers, I have 8 possible slots to finish in, first through eighth:<\/span><\/p>\n

_____<\/span> _____<\/span> _____<\/span> _____<\/span> _____<\/span> _____<\/span> _____<\/span> _____<\/span><\/p>\n

I have 8 choices for who finishes first (which was, let\u2019s be real, always Yoshi, if you were any good at that game). After first place is filled, how many choices do I have left for second? Seven. For third? Six. Then five, four, three, two, one.<\/span><\/p>\n

So multiplying my number of options together, I end up with 8*7*6*5*4*3*2*1 = 8! options. This is why you see so many factorials in combinatorics problems: we\u2019re multiplying options together as we lose an option at each step.<\/span><\/p>\n

In general, if I have n items that I am arranging, there are n! total outcomes. <\/span><\/p>\n

USE THIS WHEN: You\u2019re asked to figure out how many ways there are to arrange items\u2014that is, when <\/span>order matters<\/span><\/i>.<\/span><\/p>\n

(You can also use this to help guess on harder questions. If you have n items, you can never have an answer bigger than n! It is the absolute maximum).<\/span><\/p>\n

And Then We Get Trickier…<\/b><\/h4>\n

\u201cOkay, but you know, in Mario Kart 64, the only places that really mattered were first through third. Anyone after that didn\u2019t get any points for the\u2014\u201d<\/span><\/p>\n