Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Re: Integers less than 10,000 [#permalink]
14 Oct 2009, 11:23

40

This post received KUDOS

11

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:

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.

Re: Integers less than 10,000 [#permalink]
14 Oct 2009, 19:06

2

This post received KUDOS

Expert's post

4

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:

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)

Re: Integers less than 10,000 [#permalink]
07 Apr 2010, 02:50

20

This post received KUDOS

Expert's post

9

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: \(|dd|d|dd\) 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 \(\frac{8!}{5!3!}=56\).

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 \({n+r-1}_C_{r-1}\).

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

Also see the image I found in the net about this question explaining the concept:

Attachment:

pTNfS-2e270de4ca223ec2741fa10b386c7bfe.jpg [ 63.83 KiB | Viewed 41277 times ]

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. _________________

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:

Re: Integers less than 10,000 [#permalink]
18 May 2010, 18:21

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

Re: Integers less than 10,000 [#permalink]
16 Jun 2010, 10:45

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?

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.

Re: Integers less than 10,000 [#permalink]
20 Jul 2010, 18:06

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!

Re: Integers less than 10,000 [#permalink]
12 Oct 2010, 07:35

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

Re: Integers less than 10,000 [#permalink]
12 Oct 2010, 07:46

1

This post received KUDOS

Expert's post

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? thanks in advance

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: \({n+r-1}_C_{r-1}={6+4-1}_C_{4-1}={9}C3=\frac{9!}{6!3!}\).

Low GPA MBA Acceptance Rate Analysis Many applicants worry about applying to business school if they have a low GPA. I analyzed the low GPA MBA acceptance rate at...

UNC MBA Acceptance Rate Analysis Kenan-Flagler is University of North Carolina’s business school. UNC has five programs including a full-time MBA, various executive MBAs and an online MBA...

To hop from speaker to speaker, to debate, to drink, to dinner, to a show in one night would not be possible in most places, according to MBA blogger...

Most top business schools breed their students for a career in consulting or financial services (which is slowly being displaced by high tech and entrepreneurial opportunities). Entry into...