Manhattan Question Set # 15

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richachampion: Legendary Member; Posts: 698; Joined: Tue Jul 21, 2015 12:12 am; Location: Noida, India; Thanked: 32 times; Followed by:26 members; GMAT Score:740

Manhattan Question Set # 15

by richachampion » Wed Oct 12, 2016 7:41 am

How many different 5-person teams can be formed from a group of x individuals?

(1) If there had been x + 2 individuals in the group, exactly 126 different 5-person teams could have been formed.

(2) If there had been x + 1 individuals in the group, exactly 56 different 3-person teams could have been formed.

OA: D

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Source: Beat The GMAT — Data Sufficiency | Wed Oct 12, 2016 7:41 am

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by [email protected] » Wed Oct 12, 2016 10:38 am

Hi richachampion,

While this question looks like a complex 'Combination Formula' question, we're given actual numbers to work with, so this prompt is more about understanding the concepts involved than about doing lots of calculations.

We're asked for the number of 5-person teams that can be formed from X individuals. Since we're asked for 'groups', we'll need to use the Combination Formula:

N!/(K!)(N-K)! where N is the total number of people and K is the size of the subgroup

Before we get to the two Facts, I'm going to work through a couple of simple examples...

IF... there are 5 total people, then the number of 5-person teams is 5!/(5!)(0!) = 5!/5!(1) = 1 possible team
IF... there are 6 total people, then the number of 5-person teams is 6!/(5!)(1!) = 6!/5!(1) = 6 possible teams
IF... there are 7 total people, then the number of 5-person teams is 7!/(5!)(2!) = 7!/5!(2) = 21 possible teams
Etc.

Notice how each outcome is unique (depending on the number of people we START with). That logic works 'both ways' - if we know the number of possible teams, then we can figure out the number of people we started with...

1) If there had been x + 2 individuals in the group, exactly 126 different 5-person teams could have been formed.

From the above work, we know that there will be exactly one situation in which 126 possible teams can be formed. Subtracting 2 from that number will give us the actual value of X, which we would then use to answer the question.
Fact 1 is SUFFICIENT

2) If there had been x + 1 individuals in the group, exactly 56 different 3-person teams could have been formed.

The same logic that we applied in Fact 1 will apply here (we would just need to form groups of 3 instead of groups of 5. We could find that one situation, then subtract 1 from that group to find the value of X.
Fact 2 is SUFFICIENT

Final Answer: D

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Rich

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