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Each of the integers from 0 to 9, inclusive, is written on a separate : Problem Solving (PS)
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Re: Each of the integers from 0 to 9, inclusive, is written on a separate [#permalink]
Bunuel wrote:
joynal2u wrote:
Each of the integers from 0 to 9, inclusive, is written on separrate slip of blank paper and the ten slips are dropped into a hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10?

A. 3
B. 4
C. 5
D. 6
E. 7

I am getting how to solve this problem.

Thanks in advance for your attempt


You should consider the worst case scenario: if you pick numbers 0, 1, 2, 3, 4, and 5 then no two numbers out of these 6 add up to 10.

Now, the next, 7th number whatever it'll be (6, 7, 8, or 9) will guarantee that two number WILL add up to 10. So, 7 slips must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10.

Answer: E.



Hi Bunnel,
This solution means we are actually able to decide what we are going to pick. Right?
I some how got the idea we can not see which slip we are going to pick, so my answer was 9.
If we cannot decide which one to pick then will 9 be correct?
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Re: Each of the integers from 0 to 9, inclusive, is written on a separate [#permalink]
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cumulonimbus wrote:
Bunuel wrote:
joynal2u wrote:
Each of the integers from 0 to 9, inclusive, is written on separrate slip of blank paper and the ten slips are dropped into a hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10?

A. 3
B. 4
C. 5
D. 6
E. 7

I am getting how to solve this problem.

Thanks in advance for your attempt


You should consider the worst case scenario: if you pick numbers 0, 1, 2, 3, 4, and 5 then no two numbers out of these 6 add up to 10.

Now, the next, 7th number whatever it'll be (6, 7, 8, or 9) will guarantee that two number WILL add up to 10. So, 7 slips must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10.

Answer: E.



Hi Bunnel,
This solution means we are actually able to decide what we are going to pick. Right?
I some how got the idea we can not see which slip we are going to pick, so my answer was 9.
If we cannot decide which one to pick then will 9 be correct?


Nope. The question asks "how many must be drawn to ensure that..."

To ensure that the sum will be 10 we should consider the worst possible scenario.

Similar questions to practice:
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Hope it helps.
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Re: Each of the integers from 0 to 9, inclusive, is written on a separate [#permalink]
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manish2014 wrote:
Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into a hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10?
a 3
b 4
c 5
d 6
e 7


You can make a sum of 10 by pairing these numbers:
1, 9
2, 8
3, 7
4, 6

So if you pick only one number from each of these pairs and the leftover 2 numbers: 0 and 5, you would have picked 6 numbers without any two numbers adding up to 10. When you pick the next number (7th), there will be one of the 'sum 10' pairs. So picking 7 numbers will give you one of the required pairs for sure!

Answer (E)
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Re: Each of the integers from 0 to 9, inclusive, is written on a separate [#permalink]
joynal2u wrote:
Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10?

A. 3
B. 4
C. 5
D. 6
E. 7


Just a view :

Even after drawing 7 if we pair it with 0 we will not get a 10 , Drawing 7 does not ensure that sum of ANY of the drawn slips will be 10.
Yes, we can make a 10 but we can also NOT do so.
Isn't the question asking how many we must draw to ensure (100%) that sum of ANY of the 2 drawn slips must be 10?

Maybe I am over reading the word " ANY".

How many must be withdrawn so that we CAN make a pair with a sum 10 seems , more appropriate.
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Re: Each of the integers from 0 to 9, inclusive, is written on a separate [#permalink]
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stne wrote:
joynal2u wrote:
Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10?

A. 3
B. 4
C. 5
D. 6
E. 7


Just a view :

Even after drawing 7 if we pair it with 0 we will not get a 10 , Drawing 7 does not ensure that sum of ANY of the drawn slips will be 10.
Yes, we can make a 10 but we can also NOT do so.
Isn't the question asking how many we must draw to ensure (100%) that sum of ANY of the 2 drawn slips must be 10?

Maybe I am over reading the word " ANY".

How many must be withdrawn so that we CAN make a pair with a sum 10 seems , more appropriate.


Hello

I think the question does NOT mention the word ANY. The question clearly states, "....how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10". So if we pick 7 slips, we can be completely sure that we will be able to find two slips here which will add up to '10'. And in worst case scenario, 7 slips are what are required to be drawn.
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Re: Each of the integers from 0 to 9, inclusive, is written on a separate [#permalink]
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joynal2u wrote:
Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into a hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10 ?

A. Three
B. Four
C. Five
D. Six
E. Seven

Amazing question | Sharing the method to solve backwards from answer options

Option A - Say we pick up (0, 1, 2) or (3, 4, 5) then we do not get a 10 whereas if we pick up (4, 5, 6) or (0, 1, 9) then we get a sum of 10 from two slips

Option B - Say we pick up (0, 1, 2, 3) or (2, 3, 4, 5) then we do not get a 10 whereas if we pick up (3, 4, 5, 6) or (0, 1, 2, 9) then we get a sum of 10 from two slips

Option C - Say we pick up (0, 1, 2, 3, 4) or (1, 2, 3, 4, 5) then we do not get a 10 whereas if we pick up (2, 3, 4, 5, 6) or (0, 1, 2, 3, 9) then we get a sum of 10 from two slips

Option D - Say we pick up (0, 1, 2, 3, 4, 5) then we do not get a 10 whereas if we pick up (1, 2, 3, 4, 5, 6) or (0, 1, 2, 3, 4, 8) then we get a sum of 10 from two slips

Option E - Say we pick up (0, 1, 2, 3, 4, 5, 6) or (1, 2, 3, 4, 5, 6, 7) or (4, 5, 6, 7, 8, 9, 0) or any other combination of 7 integers between 0 and 9, incl. we will get at least 1 pair of integers that sums to 10

Ans. E
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Re: Each of the integers from 0 to 9, inclusive, is written on a separate [#permalink]
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We have 10 different slips of paper with the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 listed. How many must we draw to ensure that 2 of the numbers will total 10?

The only way we can arrive at a sum of 10 from 2 slips is if we have one out of {6, 7, 8, 9}.

We have to consider the worse case scenario: if we draw all the slips aside from {6, 7, 8, 9}.

6 + 1 = 7 to ensure that 2 numbers will total 10.
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Re: Each of the integers from 0 to 9, inclusive, is written on a separate [#permalink]
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Always think of the worst case in such questions.

The worst case is when he starts from smallest number up, Thus until he reaches 5 (0,1,2,3,4,5), no 2 numbers can make 10
It is only when he picks up any other number, be it 6 or 7 or 8 or 9, he will be able to complete the sum of 2 chits as 10.

This 6+1 = 7
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Re: Each of the integers from 0 to 9, inclusive, is written on a separate [#permalink]
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Imagine you having the worse lucky picking by chance the slip of papers. Here is the sequence of the most unlucky person alive:

First drawn: number 0
Second drawn: number 1
Third drawn: number 2
Fourth drawn: number 3
Fifth drawn: number 4
Sixth drawn: number 5
Seventh drawn: number 6

4 + 6 = 10

E

I used this same technique in other similar GMAT official question. GMAT is not about tricks. It’s about patterns.
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