This month is the first time we are introducing an official GRE question as our monthly brain teaser. The below question is yet another example of how reasoning often trumps knowledge on standardized tests and how the questions can sometimes resemble brain teasers in the kind of thinking they require or reward. What follows is a “Quantitative Comparison” question, which asks you to determine whether one quantity is greater than the other, whether they are equal, or whether you do not have enough information to determine the relationship. Enjoy!
Seven one-dollar bills are to be distributed among Lucia, Gomez, and Domingo so that each person receives at least $1.
Quantity A: The number of ways to distribute the bills so that at least one person receives at least $3.
Quantity B: The total number of ways to distribute the bills.
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
Answer:
This is one of my favorite all-time GRE questions. To the uninitiated, the answer would seem, at first glance, to be choice B. How could the total number of ways to distribute the bills NOT be greater than the number of ways to distribute them that involve some restriction? But this is where knowledge of the GRE comes in handy. Choice B should feel a little too easy, too obvious: it is very likely a trap answer. At the very least we should investigate further! Another fact about the GRE is important here: Quantitative Comparison questions ask you to compare two quantities, not to calculate them. Most people who don’t immediately choose choice B try to calculate all of the ways to distribute the bills that satisfy the restriction in Quantity A. But that is not necessary! We know that Quantity A cannot be bigger than Quantity B. So the only plausible alternative to answer choice B is that the quantities are equal. In other words, we don’t need to calculate the value of Quantity A, we just need to know whether it is equal to Quantity B. So then the question becomes, does the restriction in Quantity A actually impose a real restriction, or would it be the case that all of the ways to distribute the bills would have at least one person receiving at least $3? And if you simply try, you will see that there is no way to distribute the bills in such a way that one person does not end up with at least $3. So the quantities are equal and the answer is choice C!