• Veritas Prep
    Free Veritas GMAT Class
    Experience Lesson 1 Live Free

    Available with Beat the GMAT members only code

    MORE DETAILS
    Veritas Prep
  • e-gmat Exclusive Offer
    Get 300+ Practice Questions
    25 Video lessons and 6 Webinars for FREE

    Available with Beat the GMAT members only code

    MORE DETAILS
    e-gmat Exclusive Offer
  • Magoosh
    Magoosh
    Study with Magoosh GMAT prep

    Available with Beat the GMAT members only code

    MORE DETAILS
    Magoosh
  • Varsity Tutors
    Award-winning private GMAT tutoring
    Register now and save up to $200

    Available with Beat the GMAT members only code

    MORE DETAILS
    Varsity Tutors
  • Economist Test Prep
    Free Trial & Practice Exam
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    Economist Test Prep
  • Kaplan Test Prep
    Free Practice Test & Review
    How would you score if you took the GMAT

    Available with Beat the GMAT members only code

    MORE DETAILS
    Kaplan Test Prep
  • EMPOWERgmat Slider
    1 Hour Free
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    EMPOWERgmat Slider
  • PrepScholar GMAT
    5 Day FREE Trial
    Study Smarter, Not Harder

    Available with Beat the GMAT members only code

    MORE DETAILS
    PrepScholar GMAT
  • Target Test Prep
    5-Day Free Trial
    5-day free, full-access trial TTP Quant

    Available with Beat the GMAT members only code

    MORE DETAILS
    Target Test Prep

Clarification needed on combinatorics problem

This topic has 4 expert replies and 2 member replies
knight247 Legendary Member
Joined
19 Apr 2011
Posted:
504 messages
Followed by:
10 members
Upvotes:
114
Target GMAT Score:
800

Clarification needed on combinatorics problem

Post Fri Nov 21, 2014 10:03 am
In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts?

OA 16! ÷ (4!)^4


I already got the answer by doing 16C4 * 12C4 * 8C4 * 4C4

My question is, since the four children are NOT identical, shouldn't the above calculation also have a 4C1*3C1*2C1 in there? Considering the four children are NOT identical, we would need to pick one kid each time we need to assign a kid an assortment of four gifts, don't we?

Detailed explanations would be appreciated. Many thanks in advance.

  • +1 Upvote Post
  • Quote
  • Flag
Dutta Newbie | Next Rank: 10 Posts Default Avatar
Joined
31 Jan 2016
Posted:
2 messages
Post Mon Feb 08, 2016 9:54 am
Hey Brent,

Since the 4 kids are not identical should we not consider selecting as to who receives the 1st set of 4 gifts and who the 2nd and so on. So shouldn't the ans be multiplied by a 4!?


Brent@GMATPrepNow wrote:
knight247 wrote:
In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts?
If we take the task of distributing the 16 gifts and break it into [b]stages, we can see that we need not perform the additional calculations you suggest.

Let's say the children are named A, B, C, and D

Stage 1: Select 4 gifts to give to child A
Since the order in which we select the 4 gifts does not matter, we can use combinations.
We can select 4 gifts from 16 gifts in 16C4 ways (= 16!/(4!)(12!))
So, we can complete stage 1 in 16!/(4!)(12!) ways

Stage 2: select 4 gifts to give to child B
There are now 12 gifts remaining
Since the order in which we select the 4 gifts does not matter, we can use combinations.
We can select 4 gifts from 12 gifts in 12C4 ways (= 12!/(4!)(8!))
So, we can complete stage 2 in 12!/(4!)(8!) ways


Stage 3: select 4 gifts to give to child C
There are now 8 gifts remaining
We can select 4 gifts from 8 gifts in 8C4 ways (= 8!/(4!)(4!))
So, we can complete stage 3 in 8!/(4!)(4!) ways

Stage 4: select 4 gifts to give to child C
There are now 4 gifts remaining
NOTE: There's only 1 way to select 4 gifts from 4 gifts, but if we want the answer to look like the official answer, let's do the following:
We can select 4 gifts from 4 gifts in 4C4 ways (= 4!/4!)
So, we can complete stage 4 in 4!/4! ways

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus distribute all 16 gifts) in [16!/(4!)(12!)][12!/(4!)(8!)][8!/(4!)(4!)][4!/4!] ways

A BUNCH of terms cancel out to give us (= 16!/(4!)⁴)

--------------------------

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video: http://www.gmatprepnow.com/module/gmat-counting?id=775

Then you can try solving the following questions:

EASY
- http://www.beatthegmat.com/what-should-be-the-answer-t267256.html
- http://www.beatthegmat.com/counting-problem-company-recruitment-t244302.html
- http://www.beatthegmat.com/picking-a-5-digit-code-with-an-odd-middle-digit-t273110.html
- http://www.beatthegmat.com/permutation-combination-simple-one-t257412.html
- http://www.beatthegmat.com/simple-one-t270061.html
- http://www.beatthegmat.com/mouse-pellets-t274303.html


MEDIUM
- http://www.beatthegmat.com/combinatorics-solution-explanation-t273194.html
- http://www.beatthegmat.com/arabian-horses-good-one-t150703.html
- http://www.beatthegmat.com/sub-sets-probability-t273337.html
- http://www.beatthegmat.com/combinatorics-problem-t273180.html
- http://www.beatthegmat.com/digits-numbers-t270127.html
- http://www.beatthegmat.com/doubt-on-separator-method-t271047.html
- http://www.beatthegmat.com/combinatorics-problem-t267079.html


DIFFICULT
- http://www.beatthegmat.com/wonderful-p-c-ques-t271001.html
- http://www.beatthegmat.com/ps-counting-t273659.html
- http://www.beatthegmat.com/permutation-and-combination-t273915.html
- http://www.beatthegmat.com/please-solve-this-real-gmat-quant-question-t271499.html
- http://www.beatthegmat.com/no-two-ladies-sit-together-t275661.html
- http://www.beatthegmat.com/laniera-s-construction-company-is-offering-home-buyers-a-wi-t215764.html

Cheers,
Brent

  • +1 Upvote Post
  • Quote
  • Flag

GMAT/MBA Expert

Post Mon Feb 08, 2016 1:50 pm
Dutta wrote:
Hey Brent,

Since the 4 kids are not identical should we not consider selecting as to who receives the 1st set of 4 gifts and who the 2nd and so on. So shouldn't the ans be multiplied by a 4!?
We have already accounted for the children being non-identical.
At each stage, we give gifts to a particular child (child A, B, C and D)

Cheers,
Brent

_________________
Brent Hanneson – Founder of GMATPrepNow.com
Use our video course along with Beat The GMAT's free 60-Day Study Guide

Check out the online reviews of our course
Come see all of our free resources

  • +1 Upvote Post
  • Quote
  • Flag
GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months!

GMAT/MBA Expert

Post Thu Oct 15, 2015 11:14 am
prada wrote:
Hey Mitch on your last line I believe you mean 12C4 and not 12C5?
Thanks for pointing out the typo.
I've amended my solution accordingly.

_________________
Mitch Hunt
GMAT Private Tutor
GMATGuruNY@gmail.com
If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.
Available for tutoring in NYC and long-distance.
For more information, please email me at GMATGuruNY@gmail.com.

  • +1 Upvote Post
  • Quote
  • Flag
Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.

Top Member

prada Senior | Next Rank: 100 Posts Default Avatar
Joined
08 Dec 2010
Posted:
64 messages
Upvotes:
1
Most Responsive Member
Post Thu Oct 15, 2015 11:01 am
GMATGuruNY wrote:
knight247 wrote:
In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts?

I already got the answer by doing 16C4 * 12C4 * 8C4 * 4C4.
Your solution is correct.
Let the four children be Adam, Bobby, Cindy and David.
From 16 gifts, the number of ways to choose 4 to give to Adam = 16C4.
From the remaining 12 gifts, the number of ways to choose 4 to give to Bobby = 12C4.
From the remaining 8 gifts, the number of ways to choose 4 to give to Cindy = 8C4.
From the remaining 4 gifts, the number of ways to choose 4 to give to David = 4C4.
To combine these options, we multiply:
16C4 * 12C5 * 8C4 * 4C4.
Hey Mitch on your last line I believe you mean 12C4 and not 12C5?

  • +1 Upvote Post
  • Quote
  • Flag

GMAT/MBA Expert

Post Fri Nov 21, 2014 10:17 am
knight247 wrote:
In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts?
If we take the task of distributing the 16 gifts and break it into [b]stages, we can see that we need not perform the additional calculations you suggest.

Let's say the children are named A, B, C, and D

Stage 1: Select 4 gifts to give to child A
Since the order in which we select the 4 gifts does not matter, we can use combinations.
We can select 4 gifts from 16 gifts in 16C4 ways (= 16!/(4!)(12!))
So, we can complete stage 1 in 16!/(4!)(12!) ways

Stage 2: select 4 gifts to give to child B
There are now 12 gifts remaining
Since the order in which we select the 4 gifts does not matter, we can use combinations.
We can select 4 gifts from 12 gifts in 12C4 ways (= 12!/(4!)(8!))
So, we can complete stage 2 in 12!/(4!)(8!) ways


Stage 3: select 4 gifts to give to child C
There are now 8 gifts remaining
We can select 4 gifts from 8 gifts in 8C4 ways (= 8!/(4!)(4!))
So, we can complete stage 3 in 8!/(4!)(4!) ways

Stage 4: select 4 gifts to give to child C
There are now 4 gifts remaining
NOTE: There's only 1 way to select 4 gifts from 4 gifts, but if we want the answer to look like the official answer, let's do the following:
We can select 4 gifts from 4 gifts in 4C4 ways (= 4!/4!)
So, we can complete stage 4 in 4!/4! ways

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus distribute all 16 gifts) in [16!/(4!)(12!)][12!/(4!)(8!)][8!/(4!)(4!)][4!/4!] ways

A BUNCH of terms cancel out to give us (= 16!/(4!)⁴)

--------------------------

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video: http://www.gmatprepnow.com/module/gmat-counting?id=775

Then you can try solving the following questions:

EASY
- http://www.beatthegmat.com/what-should-be-the-answer-t267256.html
- http://www.beatthegmat.com/counting-problem-company-recruitment-t244302.html
- http://www.beatthegmat.com/picking-a-5-digit-code-with-an-odd-middle-digit-t273110.html
- http://www.beatthegmat.com/permutation-combination-simple-one-t257412.html
- http://www.beatthegmat.com/simple-one-t270061.html
- http://www.beatthegmat.com/mouse-pellets-t274303.html


MEDIUM
- http://www.beatthegmat.com/combinatorics-solution-explanation-t273194.html
- http://www.beatthegmat.com/arabian-horses-good-one-t150703.html
- http://www.beatthegmat.com/sub-sets-probability-t273337.html
- http://www.beatthegmat.com/combinatorics-problem-t273180.html
- http://www.beatthegmat.com/digits-numbers-t270127.html
- http://www.beatthegmat.com/doubt-on-separator-method-t271047.html
- http://www.beatthegmat.com/combinatorics-problem-t267079.html


DIFFICULT
- http://www.beatthegmat.com/wonderful-p-c-ques-t271001.html
- http://www.beatthegmat.com/ps-counting-t273659.html
- http://www.beatthegmat.com/permutation-and-combination-t273915.html
- http://www.beatthegmat.com/please-solve-this-real-gmat-quant-question-t271499.html
- http://www.beatthegmat.com/no-two-ladies-sit-together-t275661.html
- http://www.beatthegmat.com/laniera-s-construction-company-is-offering-home-buyers-a-wi-t215764.html

Cheers,
Brent

_________________
Brent Hanneson – Founder of GMATPrepNow.com
Use our video course along with Beat The GMAT's free 60-Day Study Guide

Check out the online reviews of our course
Come see all of our free resources

  • +1 Upvote Post
  • Quote
  • Flag
Thanked by: knight247
GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months!

GMAT/MBA Expert

Post Fri Nov 21, 2014 10:14 am
knight247 wrote:
In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts?

I already got the answer by doing 16C4 * 12C4 * 8C4 * 4C4.
Your solution is correct.
Let the four children be Adam, Bobby, Cindy and David.
From 16 gifts, the number of ways to choose 4 to give to Adam = 16C4.
From the remaining 12 gifts, the number of ways to choose 4 to give to Bobby = 12C4.
From the remaining 8 gifts, the number of ways to choose 4 to give to Cindy = 8C4.
From the remaining 4 gifts, the number of ways to choose 4 to give to David = 4C4.
To combine these options, we multiply:
16C4 * 12C4 * 8C4 * 4C4.

_________________
Mitch Hunt
GMAT Private Tutor
GMATGuruNY@gmail.com
If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.
Available for tutoring in NYC and long-distance.
For more information, please email me at GMATGuruNY@gmail.com.



Last edited by GMATGuruNY on Thu Oct 15, 2015 11:11 am; edited 1 time in total

  • +1 Upvote Post
  • Quote
  • Flag
Thanked by: knight247
Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.

Best Conversation Starters

1 Roland2rule 170 topics
2 lheiannie07 110 topics
3 ardz24 56 topics
4 LUANDATO 50 topics
5 swerve 48 topics
See More Top Beat The GMAT Members...

Most Active Experts

1 image description Brent@GMATPrepNow

GMAT Prep Now Teacher

155 posts
2 image description Scott@TargetTestPrep

Target Test Prep

122 posts
3 image description GMATGuruNY

The Princeton Review Teacher

121 posts
4 image description Rich.C@EMPOWERgma...

EMPOWERgmat

110 posts
5 image description DavidG@VeritasPrep

Veritas Prep

81 posts
See More Top Beat The GMAT Experts