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# At least one pencil is distributed to each of the 19 students. Did at

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At least one pencil is distributed to each of the 19 students. Did at [#permalink]
ziyuen wrote:
At least one pencil is distributed to each of the 19 students. Did at least two students receive the same number of pencils?

1) The number of pencils distributed to each student is less than 19.

2) The total number of pencils distributed to students is 187.

It's testing something called the Pigeonhole Principle.

The GMAT has tested that before and the design is fine.

It's simple enough, if you have n boxes and (n+1) items that go in those boxes, at least one box has at least two items.

Suppose all the students got different numbers 1, 2, 3, ..., 18

There are only 18 different positive integers less than 19

So the 19th student must duplicate one of those numbers

You can't have 19 different positive integers that are all less than 19

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Re: At least one pencil is distributed to each of the 19 students. Did at [#permalink]
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ziyuen wrote:
At least one pencil is distributed to each of the 19 students. Did at least two students receive the same number of pencils?

1) The number of pencils distributed to each student is less than 19.

2) The total number of pencils distributed to students is 187.

Responding to a pm:

No of students: 19
Each student is given atleast one pencil.

Ques: Did 2 students receive the same number of pencils?

It depends on how many pencils were distributed and how.

1) The number of pencils distributed to each student is less than 19.

There are only 18 distinct ways of giving pencils to a students (any number from 1 to 18). No one got 19 or more pencils.
But we have 19 students. So at least 2 of them will get the same number of pencils. Say first 18 students get 1, 2, 3, ...18 pencils respectively. The 19th student will get one of these 18 number of pencils only. So 2 students MUST have the same number of pencils.
Sufficient

2) The total number of pencils distributed to students is 187.

Say, we try to give a distinct number of pencils to each of the 19 students. The minimum pencils required for this will be
1 + 2+ 3 + 4 + ... + 19 = 19*20/2 = 190 pencils

But we have only 187 pencils. Since we have fewer pencils, some students will have an equal number of pencils. So 2 students MUST have the same number of pencils.
Sufficient

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Re: At least one pencil is distributed to each of the 19 students. Did at [#permalink]
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Re: At least one pencil is distributed to each of the 19 students. Did at [#permalink]
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