GMAT Tip: Counting Problems

Photograph by Michele Constantini

The GMAT Tip of the Week is a weekly column that includes advice on taking the Graduate Management Admission Test, which is required for admission to most business schools. Every week an instructor from a top test prep company will share suggestions for improving your GMAT score. This week’s tip comes from Brent Hanneson, creator of GMAT Prep Now , a Web site offering on-demand videos that teach GMAT skills.

Whenever I see a GMAT resource label its counting section as “Combinations and Permutations,” a small part of me dies. OK, that’s an exaggeration, but I am concerned about the misleading message that this label conveys. To me, it suggests that counting questions can be solved using either permutations or combinations, when this is not the case at all.

A permutation is a rearrangement of objects or values. For example, permutations of the numbers 1, 2, 3 include 3, 2, 1, as well as 2, 1, 3 and 2, 3, 1. The truth of the matter is that true permutation questions are exceedingly rare on the GMAT. In fact, if you examine the 12th and 13th editions of the Official Guide for GMAT Review, you will not find any questions that require knowledge of permutations. So, given the rarity of these question types, it’s dangerous to approach a counting question with the notion that you need only determine whether you’re dealing with a combination or a permutation and then apply one of two formulas. If you do this, you will inevitably conclude that a question is a permutation question when it is not.

My advice is to completely erase the permutation formula from your memory and approach all counting questions as follows:

First ask, “Can I use the Fundamental Counting Principle (FCP) to answer this question?” If the answer is yes, then apply the FCP. If the answer is no, look for another approach (which will likely involve an application of the combination formula, or a mixture of combinations and the FCP).

The great thing about the FCP is that it can be used to solve many GMAT counting questions. More important, it’s almost painfully easy to use. There are only three steps.

1. Take a counting task and break it into stages.
2. For each stage, determine the number of ways to accomplish that stage.
3. Multiply the number of ways to accomplish each stage. The product is your answer.

For example, imagine a problem that asks you to determine the total number of cars that can be created using three different power sources (gas, electric, hybrid), three different colors (red, black, and blue) and two different transmissions (automatic, standard). Rather than painstakingly figuring out all the combinations, simply break the problem into stages (power source, color, transmission), determine the number of possibilities for each (3, 3 and 2). Multiplying the three numbers yields the correct answer, 18.

Brent Hanneson, the creator of GMAT Prep Now,  has worked in the field of education for most of his career. He has taught courses at three different test prep companies and created comprehensive GMAT and GRE courseware packages used by the University of British Columbia and 12 other universities across North America.

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