When it comes to rings and roundabouts for permutations and combinations, if you are asked "Ten people go to a party. How many different ways can they be seated?", how come the anticlockwise and clockwise arrangements are the same? For clockwise, do we start with attendee #1 and go clockwise until we end with attendee #10, and for the anticlockwise configuration, do we start with attendee #10 and end with attendee #1?
#Roundabouts and rings problem
steep nacelle
slender folioBOT
humble crypt
The "clockwise" and "anticlockwise" you are speaking of are the same, because they don't exist in the first place. I'm just guessing at what you mean by those terms, but I think you solve the given problem as follows:
You set a circle of chairs to efficiently track already done combinations and start placing people in their seats. While this is all good and fine to do, it is by no means necessary: the chairs could be anywhere in the room, heck, they could even be flying around.
So what I'm trying to say is that you have infinetly many possibilities to make different equations, but only so many chairs to fill. When asking about "how many possibilities to be seated" there are, the question aims at the pure, abstract form of the chairs (and people).
@steep nacelle I hope this can help you.
frank hornet
The position of the chairs matters though because the answer to seeing them around a circular table is different from the answer to sitting them in a line.
What does it mean for two seatings to be the same? If they're around a circular table and they all rotate one seat in the same direction, is that the same seating? Why?
It's because they have the same neighbours
Which is also true if you take a reflection on the seating plan.
humble crypt
But it doesn't matter at all:
Make a sheet with the first line counting through all available chairs. Then asign a name/number/variable to each chair. Then swap around the names/numbers/variables to get different seatings. Nowhere in there is it necessary to look at the position of the chairs, because it just does not matter.
frank hornet
I get what you're saying, but that's just not what seating arrangements refers to in combinatorics questions.
glass walrus
When it comes to rings and roundabouts for permutations and combinations, if you...
Usually you force select one person to be seated and then you count combinations wrt the person you sat down first, usually in circular arrangements the clockwise and anticlockwise arrangements are different, but in case of certain objects (the ones i know of are garland, necklace and invertible necklace) they anticlock and clock are the same, if they are the same youd just need to do necklace =
( circle +symmetry)/2