Combinations and Permutations on the GRE
The rule of thumb is that combinations are unordered and permutations are ordered, but what does that mean? We like illustrating the difference using a social club.
- Imagine the social club has 10 different members and you’re asked, “How many groups of 3 members can you choose from the social club to make a party committee?” Would you need to do combinations or permutations in order to formulate an answer? How do you know?
- Alternatively, imagine we alter the question slightly and ask, “ An officer slate consists of a President, a Vice President, and a Treasurer. How many different officer slates can you select from the social club membership?” Is this the same question? Or is it different? Would you need to use combinations or permutations?
The questions are, in fact, quite different. So how do you apply each method on the GRE?
Solving Combinations Problems
The first question (“How many groups of 3…”) indicates that we are counting groups of 3 people, with no need to worry about which person we choose first, second, or third—i.e., order does not matter. For that reason, this is a combinations problem.
In order to answer the question, we will use the combinations formula, where n = the total number of items (10) and k = the number of items selected (3). Note that k can equal n, but can never be greater than n (we can choose all of the items in a group, but cannot choose more items than the total). Here’s the combinations formula:
Note that an exclamation point means a factorial; factorial means multiplying the number times each integer below it down to 1. For instance, 4! = 4 * 3 * 2 * 1.
Plugging our values into the equation, we get the following (make sure you reduce numbers in the extended calculations to simplify the actual multiplying you have to do):
Therefore, we could choose 120 different groups of 3 party committees.
Solving Permutation Problems
The second question asks, “How many different ways can you select a 3-person slate of officers?” This wording tells us that we should track each selection independently, rather than by groups of 3. For example, selecting Nick as President, then Kim as Vice President, then Priyanka as Treasurer would not be the same as selecting Kim as President, then Priyanka as Vice President, then Nick as treasurer, which would not be the same as selecting Kim as President, then Nick as Vice President, then Priyanka as Treasurer, and so on—i.e., order matters. For that reason, this is a permutations problem. In order to answer this question, we will use the following permutations formula:
As you can see, the denominator is the point of difference between the combinations and permutations formulas. For any values of n and k, the number of combinations we can form will always be smaller than the number of permutations we can form. This problem is no exception. Plugging our values into the equation, then reducing as much as possible, we get:
So, when order matters and we track each selection differently, there are 720 different ways we can choose 3 officers.
Pay Attention to Language
The GRE test-makers create challenging problems by using subtle language to indicate whether you should use a combination or permutation formula to answer the question at hand. Combination questions will indicate that you need to form groups or sets; permutation questions will have words or phrases that indicate order, such as “first, second, third” or “how many different ways.” Some really tricky problems can offer up a mixture of the two.
As the old adage says, “practice makes perfect”—the more of these problems you do (and the more corresponding explanations you read), the better prepared you will be to ace combinations and permutations questions on GRE Test Day.