Simplifying Permutations and Combinations on the GMAT

by on June 24th, 2015

math_FormulaAlthough they appear less frequently than questions involving algebra or number properties, permutation and combination questions on the GMAT can be tricky for even the most savvy of test takers. As complicated as these questions can be to understand, they can also be quickly simplified. Before we go any further, let’s review the formulas for permutations and combinations (n represents the number of options and r represents the number of objects at a time):

Combination formula (nCk):
n! / k!(n-k)!

Permutation formula (nPk):
n! / (n-k)!

The formulas themselves are fairly straightforward. However, the most difficult part of solving a permutation or combination problem on the GMAT is determining which formula to use. We can simplify this by asking ourselves one question: do I need to consider the order of things?

If the answer is yes, you know you’re dealing with a permutation. Permutations will ask you to solve problems in which the order of things matters. For instance, you might be asked to find the number of possibilities for a dinner menu or a baseball lineup. On the other hand, if the order doesn’t matter, you’re only being asked to find the number of possible combinations of a group of things.

While this seems like an easy enough question to answer, take a look at the example below and see if you can identify whether you should use the permutation or combination formula (try your best not to scroll down first!).

A youth volleyball coach needs to select 6 girls from a group of 8 during a tryout. How many different combinations of teams can the coach select?

Let’s break this down into easy steps:

Step 1: Does the order of the players change the group?

In this case, since the order of each of these sub-groups will not have a direct impact on its arrangement, it’s clear that we’re looking at a combinations problem.

Step 2: Identify n and k.

Since we need to form a group of 6 from a pool of 8, n=8 and k=6.

Step 3: Using the combinations formula, solve the problem.

8! / (8-6)!6!

8! / (2)!6!

We bring the factorial in the numerator down to ‘match’ the factorial in the denominator, so that the two conveniently cancel out:

(8)(7)6! / (2)!6!

(8)(7) / (2)


While problems involving permutations and combinations can be tricky, you can make them far easier to digest and solve by breaking them down into these simple steps. Even though permutations and combinations won’t appear too often on the GMAT, your ability to answer these questions quickly and accurately will truly set you apart from the pack. Happy studying!

This post appeared first on the Economist GMAT Tutor blog.

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