The Hardest GRE Math Question I've Ever Seen
8/23/2019
(My 2 main options to help you with your GRE prep are HERE.)
You're gonna get this hard GRE math question wrong.
Ok - I admit it - I was just trying to get your attention with that line. But seriously - I've never had a GRE tutoring student or class student who answered the following actual GRE question correctly. It's the hardest GRE math question in the Official Guide, and I couldn't figure it out the first time I saw it, either. :)
Let's check it out and then see why you should care about GRE math challenge questions:
A certain bookcase contains ten books. Four are biographies and six are novels. A student must select four books; however, two or more of her selections must be biographies. How many different ways are there for the student to select four books?
(The solution is at the end of the article.)
P.S. - did you know I have a GRE math practice app?
The reason I think this question is so hard is that, although it is testing concepts with which students may be familiar, those concepts are presented differently than they'd be in math classes or textbooks. Welcome to the GRE - knowledge of concepts isn't sufficient. Mastery of those concepts is what's required - and lots of experience with official ETS GRE math questions.
Why Hard GRE Math Questions Help You
Before I explain the question (please try it if you're so inclined), let me explain why hard questions matter. Here are three ideas:
1. They force a student to think. In doing so, the student builds math ability.
2. If a student can solve them, his math confidence goes up.
3. They get students used to adapting - a crucial skill for the GRE.
It's so important that students develop a "can do" attitude towards math. They don't even need "challenge" questions. What I'd recommend is this: do not look up the answer or explanation to a question until you've truly tried your best on it. This will build your skill and confidence. If you can't get it, fine, but remember - as soon as you stop thinking and look up the answer, you're missing a chance to strengthen your math muscles and confidence.
Pro Tip: After you read a new question, give yourself about 10 seconds to just think about it. Do not be in a rush to begin. This will provide a valuable opportunity to notice the big picture / logic of the question, which may make the question much easier. Practice this so you make it a habit and so that you do it on the real test!
An important note: although I do like books like the Manhattan Prep 5-lb GRE practice book for repetition with math concepts, it's much more important to eventually gain as much experience as possible solving ETS-written math questions, because ETS writes the GRE, and because other companies' questions don't imitate ETS's logical reasoning and shortcut components to math questions.
I know, I know. Many of you will go through the ETS practice books and tests and not feel as though too many of the questions are difficult. That's ok. I still want you to work on those materials for three main reasons:
- You'll get used to finding the logical shortcuts that are baked into many ETS questions
- You'll get used to the attention to detail you'll need to make sure you don't misread questions or make careless mistakes
- You'll get used to ETS's wordiness!
I repeat: other companies' quant questions will NOT do an adequate job at preparing you for these skills. So no matter what else you do, finish all the ETS material.
GRE Math Question Solution
There are three scenarios to consider here. In the first, the student picks all four biographies. There is only one way to do this.
In the second, the student picks three biographies. To find out how many ways the student can select 3 from a group of 4, you can use the groups formula**:
(4 * 3 * 2 * 1)
-------------------
(3 * 2 * 1) (1)
This reduces to four ways. Or, you could write out the possibilities, using A, B, C, and D to represent each biography.
The next step in this second scenario is to figure out how many ways there are to choose one novel from six (since you're choosing three biographies, you must also pick a novel). Logically, there are six ways to pick one thing from a group of six.
Finally (for this scenario), multiply the four ways to choose three biographies by the six ways to choose a novel to get 24 additional ways.
Now... the third scenario: picking two biographies and two novels. To figure out how many ways there are to pick two biographies from four, we'll use the groups formula:
(4 * 3 * 2 * 1)
------------------
(2 * 1) (2 * 1)
Which reduces to 6 ways to pick 2 from 4. Now, we'll do the same with the novels - to find how many ways to pick two from six, it's the groups formula again:
(6 * 5 * 4 * 3 * 2 * 1)
-----------------------------
(2 * 1) (4 * 3 * 2 * 1)
Which reduces to 15: 15 ways to pick 2 from 6. Finally, multiply 15 ways to pick the two novels by 6 ways to pick the two bios and get 90 additional ways.
To get to the final answer, add the 1 initial way plus the 24 ways from the 2nd scenario plus the 90 ways from the 3rd scenario, and your answer is 115 ways.
Hope you enjoyed that! See you in a couple of weeks. What about you - do you agree that it was the hardest GRE math question you've seen?
* If you got this question right on your own, please tell me and I will be impressed.
** the "groups" formula is used when you want to find out how many groups of a certain number you can make from a larger group. For example, if you wanted to figure out how many groups of 3 you could make out of a group of 8. The arrangement of the group doesn't matter in this kind of problem (or we'd have to use a different method): the group ABC is the same as CBA or BAC. I'd define the formula as:
(big number)!
-------------------------------------------------------------
(little number)! (big number - little number)!
Where To Find More Difficult GRE Math Questions
People with high score goals often end up looking for harder GRE quantitative practice questions. Below are my top recommendations.
Note: This page contains Amazon affiliate links and I earn a commission if you purchase things through them. However, any commission I earn comes at no additional cost to you, and you pay nothing extra. My recommendations are based on deep experience with these resources, and I recommend them because they are helpful and useful, not because of the small commissions I receive if you choose to buy something.)
ETS Super Power Pack (#1 for realism)
Why buy: Contains 596 official GRE practice questions, which are worth their weight in gold. Third-party companies' practice questions and tests, to put it bluntly, suck. Not all the math questions in these books are hard, but part of becoming better at hard questions is getting lots of experience with easier ETS questions.
Trust me on this - the bigger your database of ETS math questions that you've seen, the more likely it is that you'll be able to do the random question on the screen in front of you during the real GRE!
GMAT Official Guides (#1 for cross training with math)
Why buy: If you're a high achiever and have the time for it, official GMAT material is good additional practice for quant (just don't do the GMAT's "data sufficiency" questions). GMAT questions are HARDER than GRE math questions, so they're great skill builders - just make sure you also do all available official ETS quant questions as well.
An alternative to buying these books is downloading the free GMAT Prep software and using the two tests therein. You can take those tests more than once, since they draw from a huge pool of practice questions. The website GMATclub is an excellent resource since you can sort questions by topic and difficulty level - look for the "OG" or "Quant Review" sources.
FYI: I've heard from people who switched from the GMAT to the GRE at the last minute, and a few of them crushed the GRE with almost no GRE-specific training. GMAT experience can definitely help.
ETS Powerprep GRE Tests (#1 for practice tests)
Why buy: This is as real as it gets: practice tests written by the people who create the GRE. I recommend using all five (two are free and the other three cost $39.95 each).
That's It!
I wish you the best of luck in the rest of your GRE prep! Reminder - here are the 2 ways I can help.
- Vince Kotchian