Combinations and Permutations: Ask If Order MATTERS

Here's a mistake I watch people make every single week. A counting question shows up in a GRE Quant section. The Test Taker reads it once, feels that little jolt of panic, and grabs a formula from memory. They plug in the numbers, get an answer, see that answer sitting right there in the choices, and click. WRONG. And not because the arithmetic was off. The arithmetic was perfect. They did the right math for the wrong question.

Most counting mistakes happen before any math starts. So before you touch a formula, before you even pick up your pencil to multiply, you ask ONE question: does order MATTER?

Answer that correctly and the method becomes obvious. Answer it wrong and no amount of clean arithmetic will save you.

The one question, and how to answer it

Two words first, since you'll see them everywhere. A permutation is an arrangement, where the order of the picks changes the result. A combination is a group, where the order of the picks changes nothing. That's the whole difference.

Here's how you decide which one you're facing. I call it the swap test. Write down ONE possible outcome. Then swap two of the items in it. Now ask: did I just make a NEW outcome, or the SAME one?

• A club picks a president and a vice president. Ana as president and Ben as VP. Swap them: Ben as president and Ana as VP. That's a different result. Ana would definitely notice. Order matters.
• A club picks a 2-person committee. Ana and Ben. Swap them: Ben and Ana. Same two people in the same room. Order doesn't matter.

Stop for a second and notice what that test DOESN'T rely on. It doesn't rely on keywords. It doesn't rely on you remembering which formula has the extra factorial in it. It relies on you looking at an actual outcome and thinking about it. That's the GRE in a nutshell: it's a critical thinking test that uses math as the gauge.

The method: slots first, then divide if you must

You don't need to memorize anything fancy. You need blanks on your scratch paper.

1. Draw one blank for each pick.
2. In each blank, write how many choices you have at that moment.
3. Multiply. That's your count if order matters.
4. If order does NOT matter, you overcounted. Every group got counted once for each way it could be lined up. So divide by the number of lineups of ONE group.

That last number is a factorial. The factorial of a number, written with an exclamation point, is that number times every whole number below it down to 1. So 3! = 3 × 2 × 1 = 6. A group of 3 items can be lined up 6 different ways, so if order doesn't matter, you divide your slot count by 6.

That's it. One question, some blanks, maybe one division. Let's put it to work.

Now one in the Quantitative Comparison format. In a Quantitative Comparison, you get two quantities and you decide which is bigger, whether they're equal, or whether you can't tell from the information given. The four answer choices are always the same.

The mistakes I see most often

1. Grabbing a formula before thinking. The formula isn't the problem. Picking it on autopilot is. INSTEAD: run the swap test on a real outcome FIRST, every single time, even when you're "sure."
2. Doing the counting in your head. Counting questions punish mental math more than almost anything else on the Quant section. You lose track of a slot, or you forget whether you already divided. INSTEAD: draw the blanks on paper and fill them in. It's the same reason you should sketch coordinate geometry questions even when you think you don't need to. Paper catches the mistakes your brain glides past.
3. Trusting keywords. People memorize "arrange means order matters" and "choose means it doesn't." Then the GRE asks you to "choose a first, second and third place winner," and the keyword lies to you. INSTEAD: ignore the verb. Look at the outcome and swap two items.
4. Dividing by the wrong number. You divide by the lineups of ONE group, not by the size of the whole pool. Picking 3 from 8? Divide by 3!, not 8!. INSTEAD: ask yourself "how many ways can THIS one group be lined up?" and divide by exactly that.
5. Grinding big numbers by hand. When the slots get long, cancel before you multiply. 9 × 8 × 7 ÷ 6 is just 3 × 4 × 7 once you cancel. That's the same habit as when you break everything down to primes with exponents. And yes, you have an on-screen calculator. Use it on purpose, after you've set the problem up correctly, not as a replacement for setting it up.

Why this is worth your training time

Counting questions aren't the most common questions in the Quantitative Reasoning section, but they show up, and they're among the most gettable points on the test once you have a process. Each Quant section gives you 20 questions in 35 minutes. You don't have time to solve a counting question twice because you guessed the setup wrong the first time. One swap test takes a few seconds. Redoing the whole question takes minutes you don't have.

And here's the bigger idea. The trap answers on these questions aren't random. They're built from the exact mistakes above. The test writers KNOW people skip the order question. That's predictable, which means it's trainable.

Your challenge for today

Pull 10 counting questions you've already done. Before you look at any of your old work, write ONE word next to each: ORDER or GROUP. Then do the swap test on paper and check your call. Then compare to your old answers.

I'll bet that at least one question you missed before wasn't a math mistake at all. It was the wrong word. Find it, fix the habit, and that's a point you'll never give away again.