MHB How many different ways can the five of my nephews be given apples?

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The discussion revolves around distributing ten identical apples among five nephews, with the key question being whether each nephew must receive at least one apple. If some nephews can receive no apples, the total distribution methods amount to 5^10. However, if each nephew must receive at least one apple, the problem can be approached using the stars and bars theorem, resulting in C(14, 4) or 1001 ways to distribute the apples. The distinction between identical and unique apples is also noted, as it affects the calculation of distribution methods. The necessity of clarifying whether every nephew must receive an apple remains a central point of the discussion.
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Ten identical apples are to distributed among five of my nephews (A,B,C,D and E). All the ten apples are distributed. How many different ways can the five of my nephews be given apples?
 
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RTCNTC said:
Ten identical apples are to distributed among five of my nephews (A,B,C,D and E). All the ten apples are distributed. How many different ways can the five of my nephews be given apples?

Does every nephew have to receive at least apple or can some receive none? What have you tried?
 
If it is possible that some of the people get no apples, then there are 5 choices who to give the first apple to, 5 choices who to give the second apple to, ... so there are a total of 5^{10} choices. If every person must receive an apple, give one apple to each person. Then do there are 5^5 ways to distribute the other 5 apples.
 
I think this is a combination with repetition (stars and bars) question in which we are trying to place 5 -1 = 4 bars among 10 stars (apples).

So, there are C(10+4, 4) = C(14, 4) = 1001 ways.

Is this right?
 
HallsofIvy said:
If it is possible that some of the people get no apples, then there are 5 choices who to give the first apple to, 5 choices who to give the second apple to, ... so there are a total of 5^{10} choices. If every person must receive an apple, give one apple to each person. Then do there are 5^5 ways to distribute the other 5 apples.

This works if the apples are unique. If they are identical then there are many repeats in this calculation that need to be accounted for.

@RTCNTC: The first question still remains - does every person have to be given an apple?
 

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