GRE Math Practice: Foosball Combinations
An office comprised of eight employees is planning to have a foosball game. A matchup consists of four players, split into pairs. If any employee can be paired up with any other employee, then how many unique matchups result?
(A) 70
(B) 210
(C) 280
(D) 336
(E) 420
I know I said no more combinations problems but try this question, inspired by another real-life scenario. At Magoosh, we have a foosball table along with several avid players. We enjoy playing when we are not busy ‘Magooshing’.
The great thing about this problem is it requires you to think a little bit. You can’t simply use the combinations formula and hope that an answer will just pop out, as if by magic.
Solution
Sure there is 8C4 in there somewhere; after all, we have to choose four from eight. But here is where things get a little tricky. If you have Al, Bob, Chris (that’s me), and Dave playing a game, then, according to the conditions in the problem, you can break up these four as follows: Al can play with three different people.
Therefore, for any four players, there are three possible matchups. So how many ways can you choose four players from eight? That’s right, now it’s time to bring in our combinations formula (click here if you want to see a short cut for dealing with these problems). 8C4 = 70. Because we have three possible matchups from those 70 different ways of choosing four players, we get 70 x 3 = 210. Answer B.
Takeaway
An interesting takeaway from this problem – at least in my opinion – is becoming aware of the presence of math in our lives. That’s right numbers and knotty problems – like our foosball conundrum – are all around you. Time to leave a tip at a restaurant? Wondering the number of ways your friends can sit while on a road trip? Need to figure out how many miles per gallon you’re getting on that road trip? (Given that your car doesn’t have one of those newfangled dashboards that does all the work for you).
In fact, I like this idea so much, I’m going to come up with more GRE-style problems based on real-life situations. Indeed, feel free to offer any real-life based mathematical conundrums you’ve encountered.
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