BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

OG11 - DS#71

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Junior | Next Rank: 30 Posts
Posts: 10
Joined: Fri Jan 11, 2008 10:10 pm

OG11 - DS#71

by absolut_frui2233 » Tue Jan 27, 2009 4:47 am
Hey guys, this may look like a straightforward question and I may be thinking too hard about it. But could you please explain the logic behind the wording?

71. A number of people each wrote down one of the first 30 positive integers. Were any of the integers written down by more than one of the people?
1) The number of people who wrote down an integer was greater than 40.
2) the number of people who wrote down an integer was less than 70

The answer here is A

I wonder why though?? The question didn't exactly say that each person wrote DIFFERENT integers right? For statement 1), can't it be the case that there are 41 people who all wrote the number "1"?? very thin chance but can't that happen?

Would appreciate any clarifications :)
Top
Source: — Data Sufficiency |
User avatar
Site Admin
Posts: 2567
Joined: Thu Jan 01, 2009 10:05 am
Thanked: 712 times
Followed by:550 members
GMAT Score:770

by DanaJ » Tue Jan 27, 2009 5:18 am
Hey there, we're not looking for the number of integers selected by the group, we're actually trying to find out if there were any integers repeated at all.

1. There are only 30 integers form 1 to 30, so the maximum number of people who could have written different integers is 30. Since there were more than 30 people (actually, even more than 40) who wrote down numbers, this means that at least 10 of the positive integers must have been repeated. So 1 is sufficient.

2. is not sufficient because you have no way of telling if you had, let's say, 15 people or 31 people. If you had 15 people, than each might have picked a different number from the 30 available. But if you have 31 people, 30 might have picked a different number, but the 31st person must have picked a number that was previously chosen. So 2 is not sufficient.

Answer A.
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 2623
Joined: Mon Jun 02, 2008 3:17 am
Location: Montreal
Thanked: 1090 times
Followed by:355 members
GMAT Score:780

by Ian Stewart » Tue Jan 27, 2009 4:56 pm
DanaJ wrote:Hey there, we're not looking for the number of integers selected by the group, we're actually trying to find out if there were any integers repeated at all.

1. There are only 30 integers form 1 to 30, so the maximum number of people who could have written different integers is 30. Since there were more than 30 people (actually, even more than 40) who wrote down numbers, this means that at least 10 of the positive integers must have been repeated. So 1 is sufficient.
You have the right answer here, but the logic in the highlighted portion above isn't quite right. We might have a thousand people who all wrote down the number '25', for example; it's possible that only one number was repeated, and that the rest weren't chosen at all.

edit- actually, I may have misunderstood your meaning. If you're saying that at least ten people must have chosen numbers that matched numbers that were already chosen, then you're entirely correct.

This question is based on the 'Pigeonhole Principle', one of the fundamental principles of mathematical counting. The Pigeonhole Principle is a very simple idea, but has some interesting consequences, for example, an example provided by the wikipedia article : "there must be at least two people in London with the same number of hairs on their heads."

It won't be helpful for the GMAT, but if anyone's interested:

en.wikipedia.org/wiki/Pigeonhole_principle
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com
Top
Junior | Next Rank: 30 Posts
Posts: 10
Joined: Fri Jan 11, 2008 10:10 pm

by absolut_frui2233 » Tue Jan 27, 2009 11:41 pm
Ian Stewart wrote:
DanaJ wrote:Hey there, we're not looking for the number of integers selected by the group, we're actually trying to find out if there were any integers repeated at all.

1. There are only 30 integers form 1 to 30, so the maximum number of people who could have written different integers is 30. Since there were more than 30 people (actually, even more than 40) who wrote down numbers, this means that at least 10 of the positive integers must have been repeated. So 1 is sufficient.
You have the right answer here, but the logic in the highlighted portion above isn't quite right. We might have a thousand people who all wrote down the number '25', for example; it's possible that only one number was repeated, and that the rest weren't chosen at all.

edit- actually, I may have misunderstood your meaning. If you're saying that at least ten people must have chosen numbers that matched numbers that were already chosen, then you're entirely correct.

This question is based on the 'Pigeonhole Principle', one of the fundamental principles of mathematical counting. The Pigeonhole Principle is a very simple idea, but has some interesting consequences, for example, an example provided by the wikipedia article : "there must be at least two people in London with the same number of hairs on their heads."

It won't be helpful for the GMAT, but if anyone's interested:

en.wikipedia.org/wiki/Pigeonhole_principle
Thanks for the answer guys...I read through the pigeonhole article and still didn't get the logic behind it though...can u please explain again why it ISN'T possible for all 40 people to choose one single number?? there is a small probability of (1/30)^40 for that to happen isn't it
Top
User avatar
Site Admin
Posts: 2567
Joined: Thu Jan 01, 2009 10:05 am
Thanked: 712 times
Followed by:550 members
GMAT Score:770

by DanaJ » Wed Jan 28, 2009 12:18 am
Ian: yes, I was talking about the explanation you provided in the edited part of your post.
absolut_frui: I wrote this at the top of my post. Maybe you missed it:

"Hey there, we're not looking for the number of integers selected by the group, we're actually trying to find out if there were any integers repeated at all."

So if 40 people selected 1, then it is true that at least one number was repeated (actually, it was repeated 40 times!).
Top
Legendary Member
Posts: 941
Joined: Sun Dec 27, 2009 12:28 am
Thanked: 20 times
Followed by:1 members

by bhumika.k.shah » Tue Feb 09, 2010 8:55 am
The explanation in OG says

" If the number of integers to be chosen from is smaller than the number of people making the choice, then at least one of the integers has to be chosen and written down by more than one person. If the number of integers to be chosen from is the same as or greater than the number of people making the choice, it is possible that no integer will be chosen and written down more than once. "

so what my understanding is if there are 30 integers and 30 or more people then it shouldnt be repeated. Ive highlighted the part from which i assumed this.

Though each answer explanation says it should be written down by more than one person which gives an answer yes to our main question.

But then why does the first paragraph say the opposite ?

I am kinda confused . :(
Top
Junior | Next Rank: 30 Posts
Posts: 10
Joined: Mon Dec 13, 2010 8:35 am
GMAT Score:710

by jamesacorrea » Tue Jul 24, 2012 2:44 am
You know this question confused me too. For some reason I just couldn't get past the "well what does it matter if they all choose the same number?" question. I never really recognized that all they wanted to know was if just one single number would have to be repeated, which obviously, it would in the first option.

Hey there, we're not looking for the number of integers selected by the group, we're actually trying to find out if there were any integers repeated at all.
Thanks.
Top
Post new topic Post Reply
• Page 1 of 1