Master Quantitative Reasoning using the “slots method” for combinatorics.
Welcome to “Land Your Score,” a blog series in which Kaplan instructor Jennifer Land shares key insights and strategies for improving your GMAT performance on Test Day. This week, Jennifer discusses combination and permutation.
Combination and permutation
Many GMAT test-takers struggle with combinatorics. Combination and permutation have similar formulas that many test-takers can’t keep straight, and these problems are almost always of a high difficulty level. They are by their very nature some of the more challenging questions you’ll encounter in the Quantitative Reasoning section.
There is, however, a simpler way to understand combination and permutation that does not require learning the formulas. Kaplan GMAT students learn to solve using both the formulas and this handy approach, which we call the slots method. Let’s look at a sample problem from the Quantitative Reasoning section to get started:
Slots method for permutation
Amanda has 7 trophies. In how many distinct ways can she display 3 of them in her trophy case?
Because we are asked for distinct ways to display the trophies, we know this is a permutation problem. If the trophy case holds 3 trophies, you can consider that to be 3 slots: . How many trophies can she select from to fill the first slot? There are 7, so we fill in the first slot with the number of available trophies: 7 .
Since one trophy has been placed, how many choices are there for the second slot? Now there are only 6: 7 _6_ ___.
And for the third slot, she has 5 trophies from which to select: 7 6 5 .
For each of the 7 trophies available for the first slot, there are 6 possibilities available for the second slot and 5 trophies available for the third. To get the total number of possible arrangements, multiply the options for the slots: 7 x 6 x 5 = 210 possible arrangements in Amanda’s trophy case.
Slots method using factorials for combination
You can also use this approach to solve other types of combinatorics problems as well—including combination; it just takes one more step. Let’s look at a slightly different version of Amanda’s trophy problem:
Amanda has 7 trophies. How many different groups of 3 at a time can she display in her trophy case?
Now we do not need to know all of the distinct arrangements; we simply need to determine the number of groups of 3 trophies she can choose from the original 7. For permutation, where order matters, each of these is a distinct arrangement:
Trophy 1 | Trophy 2 | Trophy 3
Trophy 1 | Trophy 3 | Trophy 2
Trophy 2 | Trophy 1 | Trophy 3
Trophy 2 | Trophy 3 | Trophy 1
Trophy 3 | Trophy 1 | Trophy 2
Trophy 3 | Trophy 2 | Trophy 1
For combination, those options are merely different ways to arrange a set of 3 items. If the order doesn’t matter, then it is simply a single group rearranged 6 times. (And the number of times an item is rearranged can be found by taking the number of slots as a factorial. Here, there were 3 slots, so there are 3! = 3 x 2 x 1 = 6 duplicate sets.)
The way to find the number of combinations using the slots method is the same as finding the permutations, with one additional step at the end to remove those rearranged (duplicate) sets: divide by the number of slots factorial. In this problem, we found 210 permutations, and there were 3 slots, so we divide by 3!, or 3 * 2 * 1 = 6, and get 35 possible combinations.
Try a few practice problems on your own using the slots method for combination and permutation. You will likely find combinatorics problems much simpler to solve this way. Stay tuned—next week I will explain one of the magical properties of circles that the GMAT requires you to understand.
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