Many students feel that the GMAT is only necessary to get into business school, and otherwise serves no real purpose in their everyday lives. I, as a GMAT enthusiast (and overall math nerd), see a lot of real world applications in the concepts being tested on this exam. It’s actually somewhat surprising how often splitting the cheque at a restaurant or calculating investment returns requires me to delve into my GMAT knowledge. Such an instance just happened the other weekend, and it’s the kind of story I’d like to use to illustrate how pervasive GMAT knowledge is in daily life.
After celebrating Easter lunch, the family enjoyed dessert and spirited conversation (yelling) for a few hours. When it was time to leave, like many Mediterranean families, everyone felt the need to kiss everyone else goodbye (this is a great way to spread disease, by the way). While people were busy lining up to wish each other farewell, my GMAT brain took over. I asked myself: if there were 14 people gathered there, and everyone had to say goodbye to everyone else, how many embraces would that encompass in total?
The first idea that came to mind was 14! I quickly dismissed this idea, as this is an astronomical number. I know 10! Is about 3.5 million, so 14! Is well into the billions (87 billion and change, according to the calculator). If this were the case, we’d still be saying goodbye until 2015. However my brain instinctively went that direction for a reason. I thought about a little more.
Every person had to say goodbye to the 13 other people there. This means that I would have to say goodbye to the 13 other people. Similarly, every other person there would have to say goodbye to the 13 others as well. This leads to 14 x 13, and explains why I initially thought of factorials. However there is no need to keep multiplying by 12 and 11 and so on. 14 x 13 is essentially the answer, as every person there would get to say goodbye to everyone else. You can solve this little equation fairly quickly, especially if you know that 14 x 14 is 196 and then you drop 14 to 182.
However, 182 would not be the correct answer, because I am double counting all the goodbyes. For instance, I have counted saying goodbye to my mother, and I have also counted her saying goodbye to me. This is clearly the same event, so I should only count it once. This will be true of all the salutations, which means I must take my overall total of 182 and divide it by two. The actual answer should thus be 91.
I was confident that I had the correct answer, but surely there was a better way of solving this than going though the logic person-by-person (there is a better way, and don’t call me Shirley). In essence, this is a problem about combinatorics. I’m taking 14 individuals and making groups of 2s where the order doesn’t matter. This is a combination of 14 choose 2. Remembering that the formula for this kind of problem is n!/k!(n-k)!
Replacing the n by 14 and the k by 2, I’d get all the unordered pairings of people at my family gathering.
Simplifying the 12! That’s common to both the numerator and denominator:
Which ultimately yields 182/2 or just:
Now that we’ve solved my Easter farewell dilemma, let’s see if we can apply this same logic to actual GMAT problems:
If 10 people meet at a reunion and each person shakes hands exactly once with each of the other participants, what is the total number of handshakes?
(B) 10 * 10
(C) 10 * 9
Given that this is the same principle as the issue above, we can even see where the trap answers come into play. Answer choice A is the tempting factorial option, but it’s important to note the order of magnitude of this choice. Answer choice B essentially lets you make everyone shake hands with everyone, including the nonsensical option of shaking hands with yourself (Hello Ron, nice to meet you Ron). Answer choice C removes the self-adulation, but still does not provide the correct answer because it double counts the handshakes.
Using logic, we can validate that answer choice D is correct because everyone shakes hands with the 9 others but the handshakes are double counted. Using the mathematical formula yields
Where n is 10 and k is 2:
Which then becomes
And then simplifies to
We can also see that answer choice E would be correct if we decided that n should be 9 instead of 10 (possibly because we’re on a wicked bender). As is often the case, the GMAT test makers do not pick four arbitrary values for their other four answers, but rather choices you could realistically get to on this problem. Be wary not to fall into the traps laid out for you by combining your knowledge of the formula with your use of logic.
One takeaway I really like from this question is that this is the type of problem you can solve in 30 seconds or less (like a really fast pizza). If you understand what is going on here, it’s really just a question of taking n, multiplying it by n-1 and dividing by 2. This applies to any round-robin style tournament, which is the colloquial term for a tournament where everyone meets every other team.
As such, if you have a round-robin tournament of 16 teams, then you’ll just have 120 games to watch over (16 x 15 / 2). This might help to explain why the March Madness tournament is done as an elimination tournament, because otherwise the 64 teams would be playing well into the summer. Having certain question types that you understand ahead of time will help you succeed on the GMAT, and hopefully at your next gathering you’ll have good news to share with everyone before saying your goodbyes.
Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.