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

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Discussion Overview

The discussion revolves around the problem of distributing ten identical apples among five nephews (A, B, C, D, and E). Participants explore different methods of counting the distributions, considering whether each nephew must receive at least one apple or if some can receive none.

Discussion Character

  • Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant asks whether every nephew must receive at least one apple or if some can receive none, indicating a need for clarification on the problem's constraints.
  • Another participant proposes that if some nephews can receive no apples, the total number of distributions is calculated as 5^{10}, reflecting the choices for each apple.
  • A different approach suggests that if every nephew must receive at least one apple, one apple should be given to each nephew first, leading to a calculation of 5^5 for the remaining apples.
  • One participant introduces the "stars and bars" theorem, suggesting that the problem can be framed as a combination with repetition, leading to a calculation of C(14, 4) = 1001 ways, assuming the apples are identical and all can receive none.
  • Another participant reiterates the calculation of 5^{10} for the case where some can receive none, but notes that this approach may not account for the identical nature of the apples, raising concerns about overcounting.
  • There is an ongoing question about whether every person must receive an apple, which remains unresolved.

Areas of Agreement / Disagreement

Participants express differing views on the conditions of the problem, particularly regarding whether some nephews can receive no apples. There is no consensus on the correct approach or final answer.

Contextual Notes

The discussion highlights the dependence on the assumptions regarding the distribution rules, particularly whether each nephew must receive at least one apple or not. The calculations presented vary based on these assumptions.

mathdad
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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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