Welcome! Check out our free B-School Guides to learn how you compare with other applicants.
Login or Register

Clarification needed on combinatorics problem

This topic has 4 expert replies and 2 member replies
knight247 GMAT Destroyer!
Joined
19 Apr 2011
Posted:
504 messages
Followed by:
10 members
Thanked:
114 times
Target GMAT Score:
800
Clarification needed on combinatorics problem Post Fri Nov 21, 2014 10:03 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    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.

    Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums!
    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 "Thank" 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

    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.
    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 free video course along with Beat The GMAT's free 60-Day Study Guide

    Thanked by: knight247
    GMAT Prep Now's 35-hour (500+ videos) course is 100% FREE. Use our free course in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months!
    prada Rising GMAT Star Default Avatar
    Joined
    08 Dec 2010
    Posted:
    59 messages
    Thanked:
    1 times
    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?

    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 "Thank" icon.
    Available for tutoring in NYC and long-distance.
    For more information, please email me at GMATGuruNY@gmail.com.

    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.
    Dutta Just gettin' started! 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

    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 free video course along with Beat The GMAT's free 60-Day Study Guide

    GMAT Prep Now's 35-hour (500+ videos) course is 100% FREE. Use our free course in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months!

    Best Conversation Starters

    1 NandishSS 27 topics
    2 fiza gupta 20 topics
    3 Mo2men 12 topics
    4 Anaira Mitch 10 topics
    5 gabrielrc 7 topics
    See More Top Beat The GMAT Members...

    Most Active Experts

    1 image description Rich.C@EMPOWERgma...

    EMPOWERgmat

    110 posts
    2 image description GMATGuruNY

    The Princeton Review Teacher

    89 posts
    3 image description Brent@GMATPrepNow

    GMAT Prep Now Teacher

    76 posts
    4 image description Jay@ManhattanReview

    Manhattan Review

    69 posts
    5 image description DavidG@VeritasPrep

    Veritas Prep

    63 posts
    See More Top Beat The GMAT Experts