C is for Cookie

July 13, 2010 108 comments

Ten identical cookies are to be distributed among five different kids (A, B, C, D, and E). All 10 cookies are distributed. How many different ways can the five kids be given cookies?

Hints

Hint 1

You could keep track of all the different combinations: first assume A gets 10, then A gets 9, then ... but there is a faster way of doing it, that allows one to solve it in one's head, without paper, without pens!

Combinatorics, Hard, Number Theory

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Written by Anonymous on November 22, 2012. Reply

is 5544 the answer?
- Written by Steven Miller on November 22, 2012. Reply
  
  no, about 5 times less — you can email me at sjm1 AT williams.edu
Written by Jason on November 28, 2012. Reply

Does each kid get a least one cookie? Can the first kid get all ten?
- Written by Steven Miller on November 28, 2012. Reply
  
  it’s possible some kids get no cookies. it’s possible someone gets all 10. email me at sjm1 AT williams.edu to chat more
Written by Jason on November 28, 2012. Reply

Are we considering each person getting two cookies as one possible distribution? If we gave each kid two cookies and then kid A traded his cookies for kid B’s cookies, do we consider this to be another unique distribution?
- Written by Steven Miller on November 28, 2012. Reply
  
  no trades allowed, all that matters is how many you get not which
Written by Kelly Palmer on January 15, 2013. Reply

Solution please
- Written by Steven Miller on January 15, 2013. Reply
  
  Email to you bounced — email me direct at sjm1 AT williams.edu //s
Written by step by step maths on October 8, 2015. Reply

I just want to check if i’m thinking along the right lines – could we use factorials and binomial theorem here? or i suppose this doesn’t apply if each kid can get more that 1 cookie?
- Written by Steven Miller on October 10, 2015. Reply
  
  Tried to email you but it bounced — shoot me an email at sjm1 AT williams.edu
Written by Steven Miller on November 18, 2015. Reply

Paul Hanash: please email me at sjm1 AT williams.edu for the faster soln
Written by Chris on June 7, 2016. Reply

Is it just 5^10 as there is no condition that each of the five kids get a cookie? Therefore, each of the 10 cookies can be distributed five ways each.
- Written by Steven Miller on June 7, 2016. Reply
  
  no as all that matters is how many cookies you get not which ones you get
Written by Tyson on November 28, 2016. Reply

tried 5*4*3*2*1 = 120 times the 10 cookies gets 1200… anywhere close?
- Written by Steven Miller on November 28, 2016. Reply
  
  This is not correct. If you have a solution, PLEASE do not post it, but rather email me at sjm1 AT williams.edu; I will not post correct solutions as it can spoil the fun of others. If you think you got one, you may email me and I will let you know.
  - Written by Tyson on November 28, 2016. Reply
    
    sorry about that, I haven’t checked my email.
    - Written by Steven Miller on November 28, 2016. Reply
      
      I was hoping that was it — I can’t go that many hours without checking!
      
      I try not to have the solns posted as that can ruin the fun
      
      if you have any solns you want me to see, just email me at sjm1 AT williams.edu

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C is for Cookie

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