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Key GMAT Shortcuts for “Combinations” Questions

This week’s GMAT tip* concerns GMAT Combinations questions, and specifically questions that ask “How many ways…?” there are to do something. Try your hand at this sample GMAT question from the GMAT Review Official Guide, 13th Ed.:

Q: Of the three-digit integers greater than 700, how many have two digits that are equal to each other and the remaining digit different from the other two?

(A) 90
(B) 82
(C) 80
(D) 45
(E) 36

To see a detailed answer explanation and to learn key GMAT tricks for these types of questions, watch this video:

GMAT Tip of the Week: Whenever you see a GMAT problem solving question that contains the words “How many…?”, alarm bells should go off in your head and you should immediately proceed using one of two strategies:

  1. Use straight up combinatorics (see our lesson on “GMAT Probability and GMAT Combinations“)
  2. Systematically make lists and look for patterns

In fact, that’s the tips for this week: On “How many…?” GMAT questions, your best strategy is almost always to start writing out possible answers, systematically make lists, and look for patterns. Sometimes, you can literally write out every possible answer to the question using this strategy. For example, what if the question were: “How many different ice cream cones can you make out of four different flavors of ice cream and three different types of cones?” You could begin writing out the different combinations (e.g. Vanilla-Sugar Cone, Chocolate-Sugar Cone, Strawberry-Sugar Cone, Mint-Sugar Cone, Vanilla-Waffle Cone, Chocolate-Waffle Cone, etc…) and arrive at the correct answer of 12. Twelve possible combinations is a realistic number that you could literally write out on your scratch paper in under 2 minutes. Of course, you could have also applied the Fundamental Principle of Counting and arrived at the answer of 12 even more quickly. Still, Strategy #2 is useful even on problems like this.

But it is an especially useful strategy on questions like the one in the video where the answer is something large, like 90 or 82 or 80. Certainly you’re not going to have time to write out every possible answer in under two minutes, but once you start writing out some of the possible answers in a systematic way, you’ll notice patterns. You can then extrapolate those patterns, as you saw if the video, and arrive at the correct answer in a very reasonable amount of time. Seeing as there’s really no combinatorics formula for this question anyway, Strategy #2 is the best course of action on this “How many…?” GMAT question.

* Each week, one of the GMAT experts at Dominate the GMAT shares a valuable GMAT test-taking tip, strategy, trick, or content item. These tips are designed to augment your GMAT study program and provide you with additional information that will help you improve your GMAT score.