Combinatorics problem Need help

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    Combinatorics
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Discussion Overview

The discussion revolves around a combinatorics problem involving seating arrangements of delegates from 10 countries, with specific conditions regarding the proximity of certain delegates. The focus is on calculating the number of valid arrangements while adhering to the constraints provided.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a combinatorial problem and their initial calculation method, which yields a result of 544,320 combinations.
  • Another participant suggests that seating arrangements where Russia and the U.S. are next to each other could still be valid if France and England are between them.
  • A subsequent participant seeks clarification on how to incorporate the seating of Russia and the U.S. into their calculations.
  • A later reply indicates that the initial participant figured out their mistake with the help of the previous responses.
  • Another participant outlines a method to calculate the total arrangements by treating France and England as a single unit and adjusting for the seating of Russia and the U.S., arriving at the answer of 564,480 combinations.
  • The same participant points out an error in the initial calculation regarding the treatment of the proximity of Russia and the U.S. in the context of the France and England unit.

Areas of Agreement / Disagreement

Participants express differing views on the correct approach to the problem, with some clarifying and correcting earlier claims. The discussion does not reach a consensus on the initial method presented.

Contextual Notes

Participants' calculations depend on the assumptions made about the seating arrangements and the treatment of the delegates as units. There are unresolved aspects regarding the specific arrangements and the implications of the constraints on the total count.

lesdavies123
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Hi, this is the problem: Delegates from 10 countries, including Russia, France, England, and the United States, are to be seated in a row. How many different seating arrangements are possible if the French and English delegates are to be seated next to each other and the Russian and U.S. delegates are not to be next to each other.

So apparently the answer is 564 480 combinations, I come close to that, but not quite. Can anyone please correct my way of doing it which is as follows:

18 x (8!-14x(6!)) = 544 320 combinations

To justify my answer, the first 18 is for the 18 different combinations where England and France are sitting next to each other, then I tried to do the rest of the problem as if there were only 8 countries so if anyone sat next to anyone it would be 8! but - 14 (for all the different ways Russia and the US could be sitting next to each other) x 6! for all the different sitting patterns of the remaining countries! Obviously my way is wrong (unless a mistake in the answer key) so can anyone explain to me what I am doing wrong and what method would be better! Thank you very much in advance!
 
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If Russia and the US sit next to each other at the 8-party-table, it could still be a valid combination if France and England are in between in the full setup.
 
Hi, not sure if I get your answer, but do you mean if Russia and the US were only separated by France and England? How can I bring that into my equation? Thank you!
 
Hi! Figured it out thanks to your helpful answer! Thank you very much!
 
Getting their answer: Consider France and England as one unit. Ignoring the restriction on Russia and the U.S., calculate the number of permutations. That is 9!. But it is doubled because France and England can be in either order. So 2*9!. Similarly, count how many have France and England together and also have Russia and the U.S. together: 2*2*8! Subtract to get 2*9! - 2*2*8! =564480.
This is 18*8! - 16*14*6!

Your answer: 18 x (8!-14x(6!)) = 18*8! - 18*14*6!

The error: You can see that your calculation of the term -18*14*6! has the factor of 18 because you are allowing the combination of France and England to split Russia and the U.S. That is wrong. Russia and the U.S. are always together in this term. So there should be a 2*8 = 16 factor for it, not your 2*9=18 factor
 

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