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March 15th, 2019, 06:39 PM   #1
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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.
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March 15th, 2019, 07:11 PM   #2
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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.
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March 16th, 2019, 10:48 AM   #3
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Quote:
Originally Posted by Leonardox View Post
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
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March 16th, 2019, 12:21 PM   #4
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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
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March 18th, 2019, 05:54 PM   #5
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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.
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March 18th, 2019, 06:37 PM   #6
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Quote:
Originally Posted by Leonardox View Post
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)
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March 18th, 2019, 07:11 PM   #7
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stars and bars method used for indistinguishable objects into distinguishable boxes.

in our example we can tell the difference of each variety of candy.
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March 18th, 2019, 07:22 PM   #8
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Quote:
Originally Posted by Leonardox View Post
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.
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March 19th, 2019, 09:05 AM   #9
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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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