Permutation Question

[TADA]

Top Score has a question asking how many different arrangements you can have for 6 people at a round table. They state the answer is 120 or (n-1)! which in this case would be 5!. However, when I come across permutation examples such as how many ways can you arrange 6 vases on a shelf, the answer is 6! or 720. What's the difference??

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[tinman831]

When you are talking about permutations dealing with a table, you have to realize that the table is circular and there are no ends. With a shelf, there are two ends to the line.

 

[LNinlove]

Top Score has a question asking how many different arrangements you can have for 6 people at a round table. They state the answer is 120 or (n-1)! which in this case would be 5!. However, when I come across permutation examples such as how many ways can you arrange 6 vases on a shelf, the answer is 6! or 720. What's the difference??

Check this out:

(8) How many ways can Alice, Ben, Charles and Danièle arrange themselves around a square table? How many ways can they arrange themselves around a circular table?
There are 24 arrangements (24 permutations) of 4 people on 4 different sides of a square table:

ABCD BACD CABD DABC
ABDC BADC CADB DACB
ACBD BCAD CBAD DBAC
ACDB BCDA CBDA DBCA
ADBC BDAC CDAB DCAB
ADCB BDCA CDBA DCBA

For a circular table there are an infinite number of arrangements if exact position in space matters -- e.g., that Danièle can sit on the North side ot the table, the East side or at any of the infinite number of positions in between. If the only thing that matters is the relative position of each person (who is sitting next to whom), then the position of Danièle is arbitrary and there are 6 arrangement (permutations) of the other 3 people in relation to Danièle:

ABC
ACB
BAC
BCA
CAB
CBA

In this analysis there is no difference between Danièle sitting on the SouthWest side, the NorthEast side or the North by NorthWest side -- all that matters is the relative positions of the others. If Ben sits to Danièle's right, then if Alice sits to Ben's right, the arrangement will be BAC -- with Charlie to the left of Danièle.

I found it here: http://www.benbest.com/science/theodds.html#formulae

Hopefully that will be able to help you understand it a little better. I understand it as it being presented with respect to Daniele's position, rather than you setting the vases on a shelf and observing them (without counting yourself). Sorry if I don't make much sense, just got out of 3 hours of calculus. Good luck!

 

[TADA]

Check this out:

(8) How many ways can Alice, Ben, Charles and Danièle arrange themselves around a square table? How many ways can they arrange themselves around a circular table?
There are 24 arrangements (24 permutations) of 4 people on 4 different sides of a square table:

ABCD BACD CABD DABC
ABDC BADC CADB DACB
ACBD BCAD CBAD DBAC
ACDB BCDA CBDA DBCA
ADBC BDAC CDAB DCAB
ADCB BDCA CDBA DCBA

For a circular table there are an infinite number of arrangements if exact position in space matters -- e.g., that Danièle can sit on the North side ot the table, the East side or at any of the infinite number of positions in between. If the only thing that matters is the relative position of each person (who is sitting next to whom), then the position of Danièle is arbitrary and there are 6 arrangement (permutations) of the other 3 people in relation to Danièle:

ABC
ACB
BAC
BCA
CAB
CBA

In this analysis there is no difference between Danièle sitting on the SouthWest side, the NorthEast side or the North by NorthWest side -- all that matters is the relative positions of the others. If Ben sits to Danièle's right, then if Alice sits to Ben's right, the arrangement will be BAC -- with Charlie to the left of Danièle.

I found it here: http://www.benbest.com/science/theodds.html#formulae

Hopefully that will be able to help you understand it a little better. I understand it as it being presented with respect to Daniele's position, rather than you setting the vases on a shelf and observing them (without counting yourself). Sorry if I don't make much sense, just got out of 3 hours of calculus. Good luck!

Hey, that actually makes sense. My assumption was that you can take the edge of the round table and lay it out and it would convert to a shelf like problem, but as you so eloquently stated, since the table is round, there are infinite spatial arangements, so that leaves only the relative arrangements to consider. Thanks for the link too.

 

[TADA]

When you are talking about permutations dealing with a table, you have to realize that the table is circular and there are no ends. With a shelf, there are two ends to the line.

Thanks, Got it now!