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GMAT problem solving questions that contain the words “How many ways…?” may land you in a situation where you have to deal with factorials. Of course, we have useful strategies for getting certain of these types of questions right without resorting to combinatorics. However, sometimes the fastest and easiest way to answer these questions is via combinations and permutations, which may involve factorials depending on how you choose to solve them.

How will you know you’re dealing with factorials on the GMAT? You’ll see a pesky exclamation mark, which doesn’t appear to have any useful place in a math problem! (we understand +, -, /, and x, but an exclamation point?)

Alas, it does. Remember that a factorial is the product of an integer and all the integers below it. An exclamation point designates a factorial in math.

For example, 9! (read: 9 factorial) = 9*8*7*6*5*4*3*2*1.

That’s a pretty big number. In fact, it would be downright annoying to multiply those numbers out on test day without a calculator. The good news is, you likely won’t have to. You may, however, have to solve/simplify the quotient of one factorial divided by another factorial. That makes sense based on how we know to solve combination and permutation questions, doesn’t it?

Consider an example like this: How many ways are there to arrange the letters in the word STREETCAR?

Answer: 9! / (2!*2!*2!)

(Note: it’s a combination where we have 9 total letters to arrange, but three of the letters are repeated and so we have to reduce 9! by each of those redundant pairs. That’s why there are three 2! in the denominator, since there are three pairs of duplicate letters. If this is confusing, review the video lesson on “GMAT Data Analysis“).

Okay, great. But what does that work out to be? How would you identify the correct answer choice? Well, there’s a very simple way to solve this quotient, but first some explanation.

What you wouldn’t want to do is to first solve for 9! (it’s 362,800 by the way), then solve for 2!*2!*2! (8), and then divide 362,800 by 8. That’s simply too much work.

Instead, what do you notice about the denominator? If you were to write it out, do you notice that the entire denominator makes up part of the numerator?

9!               = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
2!2!2!                    2 * 1 * 2 * 1 * 2 * 1

Therefore, we can just cancel out the overlap, can’t we? So:

9!               =   9 * 8 * 7 * 6 * 5 * 4 * 3
2!2!2!                   2 * 1 * 2 * 1

The denominator is clearly 4, which cancels out the 4 in the numerator, so finally we have:

9!               =  9 * 8 * 7 * 6 * 5 * 3

Now, that’s still a pretty big number (it works out to 45,360). Fortunately you’ll see more reasonable numbers on the GMAT that you can do without a calculator. The important thing is that you understand the principle, and that a shortcut is to simply start multiplying the factorial in the numerator and STOP when you get to the first number of the factorial in the denominator.

Pop Quiz: What is 5! / 3! ?

Answer: Hopefully you quickly realized that it’s simply 5 * 4 = 20. Now that is doable.

So learn this little trick. It will save you time on test day and enable you to Dominate the GMAT!