Probability - ways in which its possible

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Discussion Overview

The discussion revolves around the problem of determining the number of ways two people can read four books simultaneously, exploring combinatorial arrangements and the concept of derangements. Participants analyze different approaches to calculating the total arrangements and express uncertainty regarding their calculations.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant outlines an initial approach to the problem, calculating options for each person's reading spots and expressing uncertainty about the remaining options for Person 2.
  • Another participant suggests that the problem is related to derangements and indicates that the correct answer is more complex than the initial guesses.
  • A third participant attempts to verify their calculations by listing all combinations and arrives at a total of 216 arrangements, while also noting the efficiency of their initial method.
  • A later reply provides a link to a resource for checking the answer, implying that there is an established method for solving the problem.

Areas of Agreement / Disagreement

Participants express differing views on the total number of arrangements, with some calculations yielding different results. The discussion remains unresolved regarding the final answer, as participants have not reached a consensus.

Contextual Notes

Participants' calculations depend on assumptions about the arrangement of books and the interpretation of the problem, which may not be fully articulated. There are also unresolved mathematical steps in the reasoning presented.

JessBrown
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So there are four books to be read by 2 people, same time to read a book each, and the question is how many different ways can they read the four books?

So I firstly took the approach that you have:
Person 1: Spot1 Spot2 Spot3 Spot4
Person 2: Spot1 Spot2 Spot3 Spot4
And then filling in how many book options there are in each spot.

So Person 1 Spot1 has 4 options.
Whereas Person2 Spot1 then has only 3 options (as one book is being read).
Therefore Person 1 Spot 2 then has 3 options left (Person1 has read one book already).
My biggest issue is in Person 2 Spot 2, I think there are 3 options, but not sure! (As Person2 has read one book, so has three remaining??) (or only 2 options?)

Then Person 1 Spot 3 has 2 opitons, Person 1 Spot4 has 1.
Person 2 Spot 3 I think has 2 remaining (or 1) and Person 2 Spot 4 has 1

Sooooo I think there are
Person 1: 4 3 2 1
Person 2: 3 3 2 1
Which results in 432 different ways in which to read the books... is this right though cos I have a feeling it may be
Person 1: 4 3 2 1
Person 2: 3 2 1 1
Which means there are only 144 options?

Thanks!
 
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This one's a nice twist on the old derangements problem. The correct answer is somewhere between the two numbers you guessed and a bit trickier to compute.

Ok, so you found that person 1 can read the books in 24 different ways. Suppose person 1 chose any order and then relabel the books 1,2,3,4 in that order. Person 2 now must choose an ordering so that 1 isn't 1st, 2 isn't 2nd etc. Now there are only 24 combinations to check which can be done by brute-force, i.e. write them all out and find the valid ones (it will be an odd number).
 
Sooooo I wrote them all down... and the answer I got was still 216, as there are nine combinations for each of the first row's orders... so 24 ways to order the first row, means 24x9 ways to order in total... the first way was so much quicker though!

eg.
1234

2143
2341
2413
3142
3412
3421
4321
4312
4123

gives nine ways for the combo of 1234 on top row, therefore there are 24 different ways for the top row, 216 in total!

anyone want to check this for me? I am fairly sure now...
 

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