*These challenge questions are great for extra practice. However, be warned: while I provide the answers, I don’t provide the explanations. If you’d like to try practice problems with explanations, check out Magoosh’s 15 GRE math practice problems and my biweekly Brain Twisters!*

Over the last few weeks, I have gone through GRE combinations and permutations. My goal is to make this formidable concept less intimidating. If you’ve watched the videos and feel comfortable with the math, I would say continue practicing but do not worry too much about this problem type. You will probably see one, though you will unlikely see two, on the actual test. So your time will better be spent on other concepts, such as work rate, sets, series, and coordinate geometry. (I will be covering these concepts soon!). Unless, that is, you are aiming for the 160+ category/the top 90% (it’s great that I can finally speak of the Revised GRE in terms of the Revised GRE scoring). The combinations and permutations I covered, while tough, may not be as difficult as a question you may see on the test, if ETS decides to make one of its most difficult questions a combinations and permutations one. Thus I have decided to devise some pretty diabolical questions. In fact, the last two are so convoluted and tricky that they wouldn’t even show up on the test. Again, this is only for those who are already comfortable with combinations and permutations, and who are going for the near perfect score. Otherwise, this could be a waste of your time (of course the less practical side of me thinks that this level of problems is fun!).

**Warming Up** An equal number of juniors and seniors are trying out for six spots on the university debating team. If the team must consist of at least four seniors, then how many different possible debating teams can result if five juniors try out?

(A) 50 (B) 55 (C) 75 (D) 100 (E) 250

**Difficulty Level: Hard** A bibliophile plans to put a total of seven books on her marble shelf. She can choose these seven books from a mixture of works from Antiquity and works on Post-modernism, of which there are seven each. If the shelf must contain at least four works from Antiquity, and one on Post-Modernism, then how many ways can he select seven books to go on the shelf?

(A) 441 (B) 1225 (C) 1666 (D) 1715 (E) 1820

**Difficulty Level: Are you kidding me?!?** An artist is planning on mixing together any number of different colors from her palette. A mixture results as long as the artist combines at least two colors. If the number of possible mixtures is less than 500, what is the greatest number of colors the artist could have in her palette? (A) 8 (B) 9 (C) 11 (D) 12 (E) 13

**Check out the answers:** 1. B 2. D 3. A

P.S. Ready to improve your GRE score? Get started today.