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!