A number of people each wrote down one of the first 30

Hi All,

This DS questions is a great example of how you can "rewrite" the question to find "the question behind the question."

We're told that an unknown number of people EACH wrote down one of the first 30 positive integers (1-30, inclusive). We're asked if ANY of the integers were written down by MORE than one person. This is a YES/NO question.

Given the 'restrictions' in this question, IF there are MORE than 30 people, then at least one of the numbers would be repeated. If there are 30 or LESS, then it's possible that a number was repeated, BUT it's also possible that NONE of the numbers were repeated. So the question is really asking...."Were there more than 30 people?"

Fact 1: The number of people....was greater than 40

This GUARANTEES that at least 1 of the numbers was written more than once, so the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT

Fact 2: The number of people....was less than 70

IF....
The total number of people was greater than 30, then the answer to the question is YES.

IF...
The total number of people is 30 or LESS, then the answer to the question COULD be YES, but it COULD be NO.
Fact 2 is INSUFFICIENT.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

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?

There are 30 integers from 1 to 30, inclusive, thus the maximum number of people possible all of them to write different integers is 30. If there are more than 30 people, then at least one of the integers has to be written by more than one person.

(1) The number of people who wrote down an integer was greater than 40 --> at least one of the integers has to be written by more than one person. Sufficient.
(2) The number of people who wrote down an integer was less than 70. Not sufficient.

Answer: A.
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How can we say that for sure? Its also possible that all the individuals wrote down 1. There is no mention that everyone has to write a unique number.

Say the number of people is 41. Each should write down one of the first 30 positive integers (1, 2, 3, ..., 30). Ask yourself, can each of them write the different integer?
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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.

I am not sure what is stopping all 40 people from writing down number 1 on their paper? Why does it has to be a sequence.. All of the 40 can just write a single number (i.e. 1) on their paper.

Yes, and in this case one of the integers (1) will be written down by more than one of the people. Thus the answer to the question will be YES. Does this make sense?
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Hi Bunnel,

The thing i am not able to understand is why not 2 ppl write the same number.In that case whether the number of ppl >40 or <70 the number that they write will be repeated.

Please assist

Hello kirtivardhan

You understand absolutely right: people from 2 statement will write the same number in any case. And we sure about this and that's why second statement Sufficient.
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Hi kirtivardhan,

The 'tricky' part about this question is that we don't know what number each individual person wrote down.

As an example....If there were only 2 people, it's POSSIBLE that they both wrote down the SAME number, but it's ALSO POSSIBLE that they wrote down two DIFFERENT numbers.

In Fact 2, we're told that there are fewer than 70 people....so there COULD be just 2 people and the above 2 results are both possible. That's why Fact 2 is INSUFFICIENT.

GMAT assassins aren't born, they're made,
Rich

(simplified wording) FOCUS: Was at least one number "chosen" more than once (among the 30 numbers available)?

(1) Sufficient: if each person (among P people) would choose a different number, it would be needed at least P numbers.
We have 30 numbers to be chosen among P > 40 people. There are at least two people who will choose the same number. (*)

(2) Insufficient: we could have only 2 people, say A and B.

> If A chose 1 and B chose 2, the answer would be NO.
> If A chose 1 and B also chose 1, the answer would be YES.


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

(*) The Pigeon Principle : https://en.wikipedia.org/wiki/Pigeonhole_principle
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Hello,
What if the question has been changed a bit?

A number of people each wrote down one of the first 5 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 2
(2) The number of people who wrote down an integer was less than 5
Is it E Bunuel?
Thanks__
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Hi Asad,

The approach that we could take with your version of the question would essentially be the same as the approach that we can take with the original form of the question (we can TEST VALUES to prove what the answer is):

We're told that an unknown number of people EACH wrote down one of the first 5 positive integers (1-5, inclusive). We're asked if ANY of the integers were written down by MORE than one person. This is a YES/NO question.

Given the 'restrictions' in this question, IF there are MORE than 5 people, then at least one of the numbers would be repeated. If there are 5 or LESS, then it's possible that a number was repeated, BUT it's also possible that NONE of the numbers were repeated.

1) The number of people who wrote down an integer was greater than 2

This information tells us that there were AT LEAST 3 people. If there were 3 people and they ALL wrote the SAME number, then the answer to the question is YES. If there were 3 people and they wrote down DIFFERENT numbers, then the answer to the question is NO.
Fact 1 is INSUFFICIENT

2) The number of people who wrote down an integer was less than 5

The examples that we used in Fact 1 can also be used in Fact 2:

This information tells us that there were NO MORE THAN 4 people. If there were 3 people and they ALL wrote the SAME number, then the answer to the question is YES. If there were 3 people and they wrote down DIFFERENT numbers, then the answer to the question is NO.
Fact 2 is INSUFFICIENT.

Combined, we know that there are either 3 people or 4 people. No additional work is required though, since we know that with 3 people, we could get a "YES" or a "NO" answer. The Final Answer would be E.

GMAT assassins aren't born, they're made,
Rich

Having read quite a few responses here, I’d like to start off by saying that the question is really about the integers and not only about the people. You could rephrase the question “Was any integer written more than once?”

The question data gives us the first 30 positive integers i.e. integers from 1 to 30.

From statement I alone, more than 40 persons wrote down an integer from the given range.

Since the number of people are more than the number of integers given (remember the number of people who wrote the integers could be 40000 also, so now it would be illogical to argue that every number hasn’t been written once and everyone has been writing the same number, say 1) , it’s obvious and very common sensical that at least one of the numbers was written more than once – it may be 1, it may be 7 it could be any of the given numbers.

And that’s precisely what the question asks – was any of integers written more than once?
Statement I alone is sufficient to answer the question with a YES. Answer options B, C and E can be eliminated.

From statement II alone, less than 70 persons wrote down an integer. Less than 70 persons could also mean 10 persons or even 5 persons.
As such, it’s not possible to say if any of the integers were written more than once.
Statement II alone is insufficient to answer the question. Answer option D can be eliminated.

The correct answer option is A.

Hope that helps!
Aravind B T

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