John's Cupcake Challenge: Finding the Perfect Distribution

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Discussion Overview

The discussion revolves around the problem of distributing 31 cupcakes among 5 students, ensuring that each student receives an odd number of cupcakes. Participants explore combinatorial methods to determine the number of ways to achieve this distribution.

Discussion Character

  • Mathematical reasoning, Technical explanation

Main Points Raised

  • John presents the problem of distributing 31 cupcakes to 5 students with the condition that each receives an odd number.
  • One participant suggests that the problem can be approached as a variation of the stars and bars problem, proposing to first add cupcakes to facilitate the distribution.
  • The method involves adding 5 cupcakes to ensure each student starts with at least one, leading to a total of 36 cupcakes, which can then be divided into stacks.
  • Another participant questions the calculation of how to divide the bars, specifically asking if it is represented as 17 choose 4 (17c4).
  • A later reply confirms the use of 17c4 as the correct combinatorial expression for the problem.

Areas of Agreement / Disagreement

Participants appear to agree on the combinatorial approach to the problem, specifically the use of the stars and bars method, but the discussion does not resolve the overall number of distributions.

Contextual Notes

The discussion does not clarify the assumptions behind the combinatorial calculations or the implications of the odd-number constraint on the distribution.

Who May Find This Useful

Individuals interested in combinatorial mathematics, particularly those studying distribution problems or the stars and bars theorem.

Monoxdifly
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John has baked 31 cupcakes for 5 different students. He wants to give them all to his students but he wants to give an odd number of cupcakes to each one. How many ways can he do this?

Brute-forcing will take about a whole day, I think. If 4 students receive 1 cupcake and the other one receive 27, that's already 4 combinations. If there are 3 1's, the other two might be 3 and 25, 5, and 23, 7 and 21, etc. Is there more efficient way?
 
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Hey Mr. Fly,

It's a variation of the stars and bars problem.
See the linked article how it works.

In this particular case we can add 5 cakes first for a total of 36.
Next we divide them in 18 stacks of 2 cakes each.
If we put 4 dividers (bars) in between them, we get 5 portions. We give them to each of the 5 students.
Oh, and before we do so, we take away 1 cake from each portion, so that each student gets an odd number.

How many ways to divide 4 bars over the 17 spaces between the stacks?
 
klaas van aarsen said:
how many ways to divide 4 bars over the 17 spaces between the stacks?

17c4?
 
Monoxdifly said:
17c4?

Yep. (Nod)
 

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