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# How many positive integers less than 10,000 are there in

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Math Expert
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How many positive integers less than 10,000 are there in [#permalink]

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13 Oct 2009, 20:37
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Difficulty:

95% (hard)

Question Stats:

44% (03:27) correct 56% (02:51) wrong based on 904 sessions

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How many positive integers less than 10,000 are there in which the sum of the digits equals 5?

(A) 31
(B) 51
(C) 56
(D) 62
(E) 93
[Reveal] Spoiler: OA

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Last edited by Bunuel on 13 Oct 2009, 21:27, edited 1 time in total.
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Re: Integers less than 10,000 [#permalink]

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14 Oct 2009, 12:23
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I believe the answer to be C: 56.

Basically, the question asks how many 4 digit numbers (including those in the form 0XXX, 00XX, and 000X) have digits which add up to 5. Think about the question this way: we know that there is a total of 5 to be spread among the 4 digits, we just have to determine the number of ways it can be spread.

Let X represent a sum of 1, and | represent a seperator between two digits. As a result, we will have 5 X's (digits add up to the 5), and 3 |'s (3 digit seperators).

So, for example:

XX|X|X|X = 2111
||XXX|XX = 0032

etc.

There are 8C3 ways to determine where to place the separators. Hence, the answer is 8C3 = 56.
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Re: Integers less than 10,000 [#permalink]

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14 Oct 2009, 20:06
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AKProdigy87 wrote:
I believe the answer to be C: 56.

Basically, the question asks how many 4 digit numbers (including those in the form 0XXX, 00XX, and 000X) have digits which add up to 5. Think about the question this way: we know that there is a total of 5 to be spread among the 4 digits, we just have to determine the number of ways it can be spread.

Let X represent a sum of 1, and | represent a seperator between two digits. As a result, we will have 5 X's (digits add up to the 5), and 3 |'s (3 digit seperators).

So, for example:

XX|X|X|X = 2111
||XXX|XX = 0032

etc.

There are 8C3 ways to determine where to place the separators. Hence, the answer is 8C3 = 56.

This is correct. Also this is the best way to solve this question. +1. (Solved exactly the same way)

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Re: Integers less than 10,000 [#permalink]

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07 Apr 2010, 03:50
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Ramsay wrote:
Sorry guys,

Could someone please explain the following:

"There are 8C3 ways to determine where to place the separators"

I'm not familiar with this shortcut/approach.

Ta

Consider this: we have 5 $$d$$'s and 3 separators $$|$$, like: $$ddddd|||$$. How many permutations (arrangements) of these symbols are possible? Total of 8 symbols (5+3=8), out of which 5 $$d$$'s and 3 $$|$$'s are identical, so $$\frac{8!}{5!3!}=56$$.

With these permutations we'll get combinations like: $$|dd|d|dd$$ this would be 3 digit number 212 OR $$|||ddddd$$ this would be single digit number 5 (smallest number less than 10,000 in which sum of digits equals 5) OR $$ddddd|||$$ this would be 4 digit number 5,000 (largest number less than 10,000 in which sum of digits equals 5)...

Basically this arrangements will give us all numbers less than 10,000 in which sum of the digits (sum of 5 d's=5) equals 5.

Hence the answer is $$\frac{8!}{5!3!}=56$$.

This can be done with direct formula as well:

The total number of ways of dividing n identical items (5 d's in our case) among r persons or objects (4 digt places in our case), each one of whom, can receive 0, 1, 2 or more items (from zero to 5 in our case) is $${n+r-1}_C_{r-1}$$.

In our case we'll get: $${n+r-1}_C_{r-1}={5+4-1}_C_{4-1}={8}C3=\frac{8!}{5!3!}=56$$

Attachment:

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Re: Integers less than 10,000 [#permalink]

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07 Apr 2010, 06:40
Fantastic explanation. +1
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Re: Integers less than 10,000 [#permalink]

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10 Apr 2010, 00:29
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amazing minds
I solved it using a lengthy process and still got it wrong. Don't know what I did wrong ...well does not matter.
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Re: Integers less than 10,000 [#permalink]

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10 Apr 2010, 03:52
Thanks Bunuel and AKProdigy +1 to you both

excellent approach.
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Re: Integers less than 10,000 [#permalink]

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13 May 2010, 16:08
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Pairs possible : 1,2,1,1 ; 1,3,1,0 ; 1,4,0,0 ; 0,1,2,2 ; 0,0,0,5 ; 0,0,2,3

which gives :
1,2,1,1 => 4!/3! = 4
0,0,0,5 =>=> 4!/3! = 4

1,3,1,0 => 4!/2! = 12
1,4,0,0 => 4!/2! = 12
0,1,2,2 => 4!/2! = 12
0,0,2,3 => 4!/2! = 12

Total = 4+4+ 12+12+12+12 = 56.

Bunnel I have solved it in this way but your methods seems to be quicker. But I couldn't understand.
Could you please explain in simple words.
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Re: Integers less than 10,000 [#permalink]

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13 May 2010, 18:14
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gurpreetsingh wrote:
Pairs possible : 1,2,1,1 ; 1,3,1,0 ; 1,4,0,0 ; 0,1,2,2 ; 0,0,0,5 ; 0,0,2,3

which gives :
1,2,1,1 => 4!/3! = 4
0,0,0,5 =>=> 4!/3! = 4

1,3,1,0 => 4!/2! = 12
1,4,0,0 => 4!/2! = 12
0,1,2,2 => 4!/2! = 12
0,0,2,3 => 4!/2! = 12

Total = 4+4+ 12+12+12+12 = 56.

Bunnel I have solved it in this way but your methods seems to be quicker. But I couldn't understand.
Could you please explain in simple words.

this approach was more natural and my first idea to solve the promblem

Then I saw the Bunnel´s and AKProdigy87´s way to solve this problem.

The idea of Bunnel and AKProdigy87 method is that the sum of the digits must equal 5, and this five can be distributed among the 4 digits and numbers are made by "ones". Again, numbers are made by "ones"! please do not thing of numbers as 2, 3, 4 or 5 digit
for example:
Quote:
XX|X|X|X = 2111
||XXX|XX = 0032

|X|XX|XX = 0|1|2|2 = 0122
X|||XXXX = 1|0|0|4 = 1004

I hope it helps
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Re: Integers less than 10,000 [#permalink]

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14 May 2010, 00:11
Thanks I got it after reading closely.

I wouldn't have cracked this way on my exam but this is quite useful way to quickly solve the question.
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Re: Integers less than 10,000 [#permalink]

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14 May 2010, 11:43
great explaination & good question
thanks Bunuel
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Re: Integers less than 10,000 [#permalink]

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17 May 2010, 11:34
Bunuel,
Great explanation!
Can you provide another example of a digit problem where we can apply this concept?
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Re: Integers less than 10,000 [#permalink]

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18 May 2010, 19:21
Quote:
Can this method be used in variations of this question?

Such as sum of 7 digits that add to equal 6?

Yes. Why not?
You mean numbers with 7 digits?
I don't know if that is going to be a question of gmat, but...
If the question was to equal 5 you can do the same to six, just add one X
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Re: Integers less than 10,000 [#permalink]

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14 Jun 2010, 09:37
bibha wrote:
HEy,
How do we know how many separators to use? I mean how did we get 8 and 3??

Hi Bibha!
Have you read the previous explanations?
8 because we have 8 elements 5 + 3
and 3 because we need a four digit number
I hope it helps
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Re: Integers less than 10,000 [#permalink]

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16 Jun 2010, 11:45
Subject: Integers less than 10,000

zestzorb wrote:
The method is called "stars and bars"

I am kinds of new to this forum and i am excited to see the happenings here. you guys are simply awesome . I have a question, will we get these kinds of complicated questions in GMAT?
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16 Jun 2010, 12:20
zestzorb wrote:
The method is called "stars and bars"

Thanks. I looked this up and am slightly confused when to use the (n-1)/(k-1) vs (n+k-1)/k or (n+k-1)/(n-1)

the / does not indicate division. Check this out link and then the link below (scroll to bottom of page for the second link). The second link uses theorem 2. Not sure I really understand the difference between the two.

http://en.wikipedia.org/wiki/Stars_and_ ... robability)

http://www.mathsisfun.com/combinatorics ... tions.html
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Re: Integers less than 10,000 [#permalink]

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20 Jul 2010, 19:06
So let's say we have the same question but we want to sum to 6 instead of 5 - then we use $$C^9_3$$ = 84. If we wanted to find numbers below 100,000 that the digits sum to 5 we would use $$C^9_4$$ = 126.

Really elegant guys. Thanks.
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Re: Integers less than 10,000 [#permalink]

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12 Oct 2010, 08:28
anilnandyala wrote:
i dont understand the concept of separators, can some one explain this

Check this:
solve-these-gmat-question-98701.html?hilit=separators#p761146
voucher-98225.html?hilit=separators#p756688

Hope it helps.
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Re: Integers less than 10,000 [#permalink]

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12 Oct 2010, 08:35
thanks Bunuel
can u explain me this by using the formulae
How many positive integers less than 10,000 are there in which the sum of the digits equals 6?
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Re: Integers less than 10,000 [#permalink]

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12 Oct 2010, 08:46
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anilnandyala wrote:
thanks Bunuel
can u explain me this by using the formulae
How many positive integers less than 10,000 are there in which the sum of the digits equals 6?

6 * (digits) and 3 ||| --> ******||| --> # of permutations of these symbols is $$\frac{9!}{6!3!}$$.

Or: The total number of ways of dividing n identical items (6 *'s in our case) among r persons or objects (4 digt places in our case), each one of whom, can receive 0, 1, 2 or more items (from zero to 6 in our case) is $${n+r-1}_C_{r-1}$$.

In our case we'll get: $${n+r-1}_C_{r-1}={6+4-1}_C_{4-1}={9}C3=\frac{9!}{6!3!}$$.

Hope it's clear.
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Re: Integers less than 10,000   [#permalink] 12 Oct 2010, 08:46

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