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March 15th, 2019, 06:39 PM  #1 
Senior Member Joined: Apr 2017 From: New York Posts: 155 Thanks: 6  how many ways..
Suppose you have got 8 varieties of candies to choose a) how many ways can you select a dozen candies? b) how many ways can you select a dozen candies with at least one of each kind? I am little bit confused since a) is selecting 12 items out of 8 = 0, but part b) is the extension of the question insisting we are selecting what do you guys think? Last edited by skipjack; March 15th, 2019 at 06:51 PM. 
March 15th, 2019, 07:11 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,968 Thanks: 2217 
Part (a) is asking for the number of ways of distributing 12 identical candies into 8 different bowls. Part (b) is the same, except for a requirement that each bowl must receive at least one candy. 
March 16th, 2019, 10:48 AM  #3  
Senior Member Joined: Feb 2010 Posts: 711 Thanks: 147  Quote:
I think: a) 50,388 b) 330  
March 16th, 2019, 12:21 PM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,549 Thanks: 1399 
a) is the problem of distributing 12 balls into 8 distinct boxes w/no restrictions. $n = \dfrac{12+81}{81} = 50388$ as mtwhs noted b) has the at least 1 per box restriction $n = \dbinom{121}{81} = 330$ also as mtwhs noted 
March 18th, 2019, 05:54 PM  #5 
Senior Member Joined: Apr 2017 From: New York Posts: 155 Thanks: 6 
Thanks for the answers but I need to know the logic of how to reach the answer, not the answer itself. Could you clarify how to reach the numbers? So I can adapt different questions by myself I appreciate. 
March 18th, 2019, 06:37 PM  #6  
Senior Member Joined: Sep 2015 From: USA Posts: 2,549 Thanks: 1399  Quote:
https://en.wikipedia.org/wiki/Stars_...combinatorics)  
March 18th, 2019, 07:11 PM  #7 
Senior Member Joined: Apr 2017 From: New York Posts: 155 Thanks: 6 
stars and bars method used for indistinguishable objects into distinguishable boxes. in our example we can tell the difference of each variety of candy. 
March 18th, 2019, 07:22 PM  #8  
Senior Member Joined: Sep 2015 From: USA Posts: 2,549 Thanks: 1399  Quote:
You are taking 12 indistinguishable candies and putting them (by type) into 8 distinguishable boxes.  
March 19th, 2019, 09:05 AM  #9 
Senior Member Joined: Apr 2017 From: New York Posts: 155 Thanks: 6 
Ok I understood this question after checking the table for this topic. And your answer completed the picture on my mind. Thanks a lot.


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