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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
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 Originally Posted by Leonardox 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?
The assumption is that you have an unlimited supply of each variety. When you walk into Dunkin Donuts and they offer 8 varieties of donut, that doesn't mean that when you buy a plain donut you can't buy another one.

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. $n = \dfrac{12+8-1}{8-1} = 50388$ as mtwhs noted b) has the at least 1 per box restriction $n = \dbinom{12-1}{8-1} = 330$ 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
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 Originally Posted by Leonardox 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.
Stars 'n' Bars

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
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 Originally Posted by Leonardox stars and bars method used for indistinguishable objects into distinguishable boxes. in our example we can tell the difference of each variety of candy.
the candy types are the boxes.

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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