Today we’ve got a fairly straightforward combinatorics GMAT problem.
John is hosting a movie night where he will show 4 different movies from among the top 12 movies of the year. He will show the movies in the order in which they are chosen. How many unique movie nights can John throw?
A. 8
B. 60
C. 495
D. 11,880
E. 154,440
If you’ve been self-prepping in a rigorous, let me review the rules sort of way. You’ll pick up that there’s orders combinations here and you might be inclined to really dive in and say “What’s my combinations formula?” “What’s my permutation formula?” “How do I know which is which?” And then plug in numbers.
While that will get you there, understand that most GMAT combinatorics problems are more about being familiar with combinatorics than any really heavy-duty math. That is because the GMAT cohort (the number of people who are taking the GMAT) are generally more familiar with Algebra or Geometry.
1. Combinatorics & The GMAT
Combinatorics, by virtue of being less known, is considered more valuable. It is scored more highly than problems of similar complexity in Algebra or Geometry. So you’re really being rewarded just for knowing basic combinatorics and in fact, most permutation/combination problems fall into this basic category. The good news here is that you can use your reasoning to solve this problem without being burdened by formal combinatorics formulas.
2. Solving The Problem
Let’s take a look at this problem. John’s having a movie night. We need to ask ourselves a series of pivot questions. How many different movies can John show first?
Well, there are 12 movies, he could show any of the 12. Leaving 11 movies to be shown second, any of 11. 10, 9. So the answer is 12x11x10x9 or 11,880. But even this math is a lot to do. Notice that by walking it through as a story, as a narrative, we don’t need to cancel out the 7 6 5 4 3 2 1. We don’t need to worry about division or anything else. We just know that there are 4 movies and each time, each step we take, there’s one less movie available. Here we have this product 12 times 11 times 10 times 9, but we don’t really want to be forced to process this. Hence, we can look for features that allow us to skip doing that heavy math.
3. Transforming The Numbers
We’ve got this really neat triangular shape in the answer choices where each answer has a different number of digits in it. 12, 11, 10, 9, we can look at and say on average each one’s about 10. The 9 and the 12 sort of compensate, but overall we’re going to have something that’s close to 10 times 10 times 10 times 10.
That is our answer should be somewhere around 10,000 or possibly a little more because we have an 11 and a 12 offset only by a 9. So what we’re looking for is something in that just above 10,000 range this prevents us from doing the math and very rapidly lets us look at those four movies, those numbers 12 11 10 9 and zero in on that 11 880 number.
4. Combinatorics GMAT Problem Form
Try it again with a similar number. Notice that you can’t do this with a hundred different movies selecting 17 of them. The math, the numbers would be too cumbersome.
The GMAT is really restricted here and you should restrict yourself to ones that are reasonable to keep processed in your head without doing heavy-duty math. Similarly, notice how this one clusters around ten, it doesn’t have to cluster around ten, but when you’re rewriting this problem think about that clustering and think about how your knowledge of common powers or how other identities can help you rapidly get to an answer because the GMAT will present you with numbers that have a neat clean way to jump from your understanding directly to the answer without all that messy math in between.
This is Mike for Apex GMAT with your problem of the day.
If you enjoyed this combinatorics GMAT problem, try more GMAT practice problems: Remainder Number Theory