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

At least one pencil is distributed to each of the 19 students. Did at least two students receive the same number of pencils?


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

This statement means that each student was distibuted from 1 - 18 pencils.

So Let us assume that each student got a different number of pencil.

So 1st got 1 pencil, 2nd got 2 pencils..till 18th got 18 pencils. Now the 19th student can get anything between 1 to 18 pencils. Which makes at least two students having the same number of pencils. Hence this statement is sufficient.

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

Using similar logic, if we try to distribute the 187 pencils as we did while analysing Statement 1, we would get

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18=171

So we are left with 187-171 = 16 pencils for the 19th student. Which makes at least two students having the same number of pencils. Hence this statement is sufficient.

Answer is D

We know that each student received at least 1 pencils and we know that there are 19 students in total.

let's say that if each student received a different number of pencils,then the total number of pencils distributed would've been 1+2+3...+19 = Sum of items * avg. = 190 total

1) We are told that the number of pencils distributed to each student is less than 19 => which means that at least two students received the same number of pencils. Sufficient
2) Sufficient. 187 is less than 190, which means that at least two students received the same number of pencils. Ans. D

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

Try this video https://youtube.com/watch?v=ROnetLvbl6M
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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

Answer (D)

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