Combinations and . . . Non-combinations

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

The truth of the matter is that true permutation questions are exceedingly rare on the GRE.

Now, for those who are unfamiliar with permutations, a permutation is an arrangement of a subset of items in a set. To be more specific:

If we have n unique objects, then we can arrange r of those objects in nPr ways, where nPr equals some formula that I still haven’t memorized even though I took several combinatorics courses in university, and I taught counting methods to high school students for 7 years.

Now it’s not that the permutation formula is too complicated to remember; it’s just that it’s unnecessary to memorize such a formula for the GRE. In my humble opinion, the permutation formula has no place in a GRE resource (even though the Official Guide covers it).

Here’s an example of a true permutation question:

Using the letters of the alphabet, how many different 3-letter words can be created if repeated letters are not permitted?

Here, we have a set of 26 letters in the alphabet, and we want to determine the number of ways of arrange 3 of those letters. So, if we still feel compelled to use permutations to the answer the question (despite my public denouncement of permutations :-), the answer would be 26P3, at which point you would have to evaluate this.

Of course you’re not going to memorize the permutation formula, because you’re going accept my premise that true permutation questions are exceedingly rare on the GRE. For the doubters out there, let’s consult the Official Guide to the GRE Revised General Test. In the Guide, there are 7 counting questions altogether. Of these 7 questions, not one is a true permutation question (although some will argue that question #6 on page 297 is a permutation question, albeit a very boring one that can be solved using an easier approach).

So, given the rarity of permutation questions, it’s dangerous to approach counting questions with the notion that all you need to do is 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 isn’t. Notice how easy it is to turn a permutation question into a non-permutation question by simply changing a word or two. For example, see what happens when we change our original question to read:

Using the letters of the alphabet, how many different 3-letter words can be created if repeated letters are permitted?

By allowing repeated letters, the question is no longer a permutation question, which means 26P3 is not the solution (the solution is 263, which we’ll cover in future posts).

The truth is that we don’t need the permutation formula to answer any counting question on the GRE (including question #6 on page 297 of the Official Guide). Instead, we can use the Fundamental Counting Principle (FCP). The FCP is easy to use and it can be used to solve the majority of counting questions on the GRE.

So, my approach with all counting questions is as follows:

  • First determine whether or not the question can be answered using the FCP
  • If the question can’t be answered using the FCP, it can probably be solved using combinations (or a combination of combinations and the FCP)

In the next series of posts, I’ll explore how we can use the FCP to solve most counting questions, and how we can use combinations to solve the rest.

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