Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 40895
Kudos [?]:
118425
[12]
, given: 11999

How many positive integers less than 10,000 are there in [#permalink]
Show Tags
13 Oct 2009, 20:37
12
This post received KUDOS
Expert's post
70
This post was BOOKMARKED
Question Stats:
45% (03:26) correct
55% (02:51) wrong based on 982 sessions
HideShow timer Statistics
Last edited by Bunuel on 13 Oct 2009, 21:27, edited 1 time in total.

Kudos [?]:
118425
[12]
, given: 11999


Manager
Joined: 11 Sep 2009
Posts: 129
Kudos [?]:
396
[60]
, given: 6

Re: Integers less than 10,000 [#permalink]
Show Tags
14 Oct 2009, 12:23
60
This post received KUDOS
20
This post was BOOKMARKED
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:
XXXXX = 2111 XXXXX = 0032
etc.
There are 8C3 ways to determine where to place the separators. Hence, the answer is 8C3 = 56.

Kudos [?]:
396
[60]
, given: 6


Math Expert
Joined: 02 Sep 2009
Posts: 40895
Kudos [?]:
118425
[4]
, given: 11999

Re: Integers less than 10,000 [#permalink]
Show Tags
14 Oct 2009, 20:06
4
This post received KUDOS
Expert's post
6
This post was BOOKMARKED
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:
XXXXX = 2111 XXXXX = 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) Answer: 56.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics

Kudos [?]:
118425
[4]
, given: 11999


Math Expert
Joined: 02 Sep 2009
Posts: 40895
Kudos [?]:
118425
[24]
, given: 11999

Re: Integers less than 10,000 [#permalink]
Show Tags
07 Apr 2010, 03:50
24
This post received KUDOS
Expert's post
18
This post was BOOKMARKED
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: \(ddddd\) 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 \(\frac{8!}{5!3!}=56\). Answer: C (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 \({n+r1}_C_{r1}\). In our case we'll get: \({n+r1}_C_{r1}={5+41}_C_{41}={8}C3=\frac{8!}{5!3!}=56\) Also see the image I found in the net about this question explaining the concept: Attachment:
pTNfS2e270de4ca223ec2741fa10b386c7bfe.jpg [ 63.83 KiB  Viewed 56746 times ]
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics

Kudos [?]:
118425
[24]
, given: 11999


Intern
Joined: 31 Mar 2010
Posts: 17
Kudos [?]:
3
[0], given: 12
Schools: Stanford GSB (waiting), Wharton (wl w/ int), INSEAD (admit)
WE 1: TopTier MC

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

Kudos [?]:
3
[0], given: 12


Manager
Joined: 07 Jan 2010
Posts: 240
Kudos [?]:
9
[1]
, given: 16

Re: Integers less than 10,000 [#permalink]
Show Tags
10 Apr 2010, 00:29
1
This post received KUDOS
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.

Kudos [?]:
9
[1]
, given: 16


Manager
Joined: 05 Mar 2010
Posts: 206
Kudos [?]:
36
[0], given: 8

Re: Integers less than 10,000 [#permalink]
Show Tags
10 Apr 2010, 03:52
Thanks Bunuel and AKProdigy +1 to you both excellent approach.
_________________
Success is my Destiny

Kudos [?]:
36
[0], given: 8


CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2771
Kudos [?]:
1812
[16]
, given: 235
Location: Malaysia
Concentration: Technology, Entrepreneurship
GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35

Re: Integers less than 10,000 [#permalink]
Show Tags
13 May 2010, 16:08
16
This post received KUDOS
5
This post was BOOKMARKED
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.
_________________
Fight for your dreams :For all those who fear from Verbal lets give it a fight
Money Saved is the Money Earned
Jo Bole So Nihaal , Sat Shri Akaal
Support GMAT Club by putting a GMAT Club badge on your blog/Facebook
GMAT Club Premium Membership  big benefits and savings
Gmat test review : http://gmatclub.com/forum/670to710alongjourneywithoutdestinationstillhappy141642.html

Kudos [?]:
1812
[16]
, given: 235


Manager
Joined: 26 Feb 2010
Posts: 78
Kudos [?]:
9
[2]
, given: 1
Location: Argentina

Re: Integers less than 10,000 [#permalink]
Show Tags
13 May 2010, 18:14
2
This post received KUDOS
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: XXXXX = 2111 XXXXX = 0032 XXXXX = 0122 = 0122 XXXXX = 1004 = 1004 I hope it helps

Kudos [?]:
9
[2]
, given: 1


CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2771
Kudos [?]:
1812
[0], given: 235
Location: Malaysia
Concentration: Technology, Entrepreneurship
GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35

Re: Integers less than 10,000 [#permalink]
Show Tags
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.
_________________
Fight for your dreams :For all those who fear from Verbal lets give it a fight
Money Saved is the Money Earned
Jo Bole So Nihaal , Sat Shri Akaal
Support GMAT Club by putting a GMAT Club badge on your blog/Facebook
GMAT Club Premium Membership  big benefits and savings
Gmat test review : http://gmatclub.com/forum/670to710alongjourneywithoutdestinationstillhappy141642.html

Kudos [?]:
1812
[0], given: 235


Manager
Joined: 16 Feb 2010
Posts: 180
Kudos [?]:
34
[0], given: 17

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

Kudos [?]:
34
[0], given: 17


Intern
Joined: 01 Dec 2008
Posts: 5
Kudos [?]:
0
[0], given: 4

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

Kudos [?]:
0
[0], given: 4


Manager
Joined: 26 Feb 2010
Posts: 78
Kudos [?]:
9
[0], given: 1
Location: Argentina

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

Kudos [?]:
9
[0], given: 1


Manager
Joined: 26 Feb 2010
Posts: 78
Kudos [?]:
9
[0], given: 1
Location: Argentina

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

Kudos [?]:
9
[0], given: 1


Intern
Joined: 11 Mar 2010
Posts: 2
Kudos [?]:
0
[0], given: 0

Re: Integers less than 10,000 [#permalink]
Show Tags
16 Jun 2010, 11:45
Subject: Integers less than 10,000 zestzorb wrote: The method is called "stars and bars"
I can't post the link. Look it up on google. 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?

Kudos [?]:
0
[0], given: 0


Intern
Affiliations: NYSSA
Joined: 07 Jun 2010
Posts: 34
Kudos [?]:
11
[0], given: 2
Location: New York City
Schools: Wharton, Stanford, MIT, NYU, Columbia, LBS, Berkeley (MFE program)
WE 1: Senior Associate  Thomson Reuters
WE 2: Analyst  TIAA CREF

zestzorb wrote: The method is called "stars and bars"
I can't post the link. Look it up on google. Thanks. I looked this up and am slightly confused when to use the (n1)/(k1) vs (n+k1)/k or (n+k1)/(n1) 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

Kudos [?]:
11
[0], given: 2


Intern
Joined: 16 Jul 2010
Posts: 18
Kudos [?]:
18
[0], given: 9

Re: Integers less than 10,000 [#permalink]
Show Tags
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.
_________________
If you find my posts useful, please award me some Kudos!

Kudos [?]:
18
[0], given: 9


Math Expert
Joined: 02 Sep 2009
Posts: 40895
Kudos [?]:
118425
[0], given: 11999

Re: Integers less than 10,000 [#permalink]
Show Tags
12 Oct 2010, 08:28

Kudos [?]:
118425
[0], given: 11999


Manager
Joined: 07 Feb 2010
Posts: 158
Kudos [?]:
679
[0], given: 101

Re: Integers less than 10,000 [#permalink]
Show Tags
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? thanks in advance

Kudos [?]:
679
[0], given: 101


Math Expert
Joined: 02 Sep 2009
Posts: 40895
Kudos [?]:
118425
[1]
, given: 11999

Re: Integers less than 10,000 [#permalink]
Show Tags
12 Oct 2010, 08:46

Kudos [?]:
118425
[1]
, given: 11999



Re: Integers less than 10,000
[#permalink]
12 Oct 2010, 08:46



Go to page
1 2 3
Next
[ 55 posts ]




