Combinations and Non-Combinations – Part I

I’ve never been a big fan of the phrase “Combinations and Permutations.” It suggests that all counting questions can be solved using either combinations or permutations, when this is not so. Compounding the problem is an infamous rule that goes something like this:

If order matters, then it’s a permutation question.

If order doesn’t matter, then it’s a combination question.

Let’s apply this rule to the following question:

A certain restaurant let’s you build your own pizza. There are 3 types of cheese, 5 different toppings, and 2 types of tomato sauce. If you must choose 1 cheese, 1 topping and 1 type of tomato sauce, how many different pizzas can you create?

Does order matter here? Well, I certainly don’t want the cheese on the bottom and the tomato sauce on the top. So, I guess order does matter, which means we can solve this using permutations, right?

No. Even though the order seems to matter, we can’t use the permutation formula to solve the question.

Aside: For those who are unfamiliar with permutations, a permutation is an arrangement of a subset of items from a set. So, if we have n unique objects, then we can arrange r of those objects in nPr ways, where nPr equals . . . some formula that would needlessly occupy valuable brain space if you were to memorize it.

Full disclosure: Although I took many Combinatorics (counting) courses in university, and taught counting methods to high school students for 7 years, I still haven’t memorized the permutation formula. Sure, I can manufacture it by applying a bit of logic, but it’s not a formula that readily comes to mind. Does my inability to recall the permutation formula prevent me from solving GMAT counting questions? Absolutely not.

The truth of the matter is that true permutation questions are exceedingly rare on the GMAT. If we consult the Official Guide for GMAT Review – 13^th edition (aka the OG13), we find a total of 5 counting questions, and not one of them is a true permutation question. Likewise, there are no questions in the OG12 or in the Quantitative Review (2^nd edition) that require the permutation formula. In fact, I don’t think I’ve ever seen a GMAT question that requires this formula. Sure, it’s possible that there exists a GMAT question that could be solved by applying the permutation formula, but I can assure you that the same question could also be solved by applying the Fundamental Counting Principle (FCP).

We’ll examine the Fundamental Counting Principle in greater detail in my next article. For this article, however, my sole purpose is to dissuade you from investing any effort in learning about permutations. If you’ve already studied permutations, I’m hoping to convince you to forget you ever heard of this concept.

Aside: I should mention that taking n objects and arranging all of them is a special kind of permutation (which can be accomplished in n! ways).  This isn’t really a scenario that most people associate with permutations, but nevertheless, this special case can be handled by applying the Fundamental Counting Principle as well.

In the long run, unlearning permutations will make it easier to solve counting questions, because, in order to apply the permutation formula correctly, you must first determine whether or not a question can be solved using permutations. This is a difficult task. To demonstrate this, see if you can identify which of these questions can be solved using permutations:

Question #1: Using the letters of the alphabet, how many different 3-letter words can be created if the first letter must be X or Q?

Question #2: Using the letters of the alphabet, how many different 3-letter words can be created if repeated letters are permitted?

How did you do?

Did you recognize that only question #2 can be solved using permutations? I hope not, because neither question can be solved using permutations.

Here’s a true permutation question:

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

Even though this question can be solved using the permutation formula, it can also be solved by applying the FCP. The FCP is easy to use, and it can be used to solve the majority of counting questions on the GMAT. So, learn the FCP and forget about permutations.

Once you learn the FCP, your approach with all counting questions will be the same.

Step 1: determine whether or not the question can be answered using the FCP

Step 2: If the question cannot be answered using the FCP, it can probably be solved using combinations (or a mix of combinations and the FCP)

Step 3: Solve

In the next series of articles, we’ll explore how the FCP works, how it can help us solve most counting questions, and how combinations can help us solve the rest.