Question

Please help. WILL MEDAL! Ricky and his six sons sat around a circular table. How many seating arrangements were possible? @mathmale @triciaal @sleepyjess @satellite73 @jim_thompson5910 @zepdrix @robtobey @aaronq @Zale101

Answer 1

wait im ronaldo lol

Answer 2

if i am not mistake, for a circle it is \((n-1)!\)

Answer 3

May you please explain more in depth? @satellite73

Answer 4

if it was not a circle, but a line, and you were arranging \(n\) people (or whatever) there would be \(n!\) ways to do it by the counting principle
n choices for the first seat, n-1 for the second, n-2 for the third and so on

Answer 5

Since you can rotate the circle however you like, this means that it does not matter who you place at the top of the circle. Let's say we always place person 1 at the top like shown here

http://mathworld.wolfram.com/CircularPermutation.html

so we go from n people to n-1 people. So we have n-1 people to arrange making (n-1)! possible permutations.

Answer 6

in a circle you would count the arrangement 1,2,3,4,5,6 the same as 2,3,4,5,6,1

Answer 7

I'm confused. :3

Answer 8

@mathmale

Answer 9

`Ricky and his six sons `
there are 1+6 = 7 people

Let's say Ricky is placed at the top of the circle as shown in the link. There would be
(n-1)! = (7-1)! = 6! = 6*5*4*3*2*1 = 720
ways to arrange the 6 sons left over

you're probably overthinking things? @okdutchman7

Answer 10

So there are 6 ways in total? @jim_thompson5910

Answer 11

I would think that there would be more than that.

Answer 12

Oh wait 720 ways.

Answer 13

Here I admit to going more by gut than by facts ( ! ), but submit that we are looking for the total number of combinations (arrangements) describing how 6 people could be positioned around a table.

Ricky sits down first. One of the 6 sons would sit next to him. (Assume we continue to seat the sons to the left of Ricky or the previous son.) After that, one of the 5 remaining sons would sit next to the previously seated son. And so on. Using this line of reasoning, 6! strikes me as correct. Ricky is always there, so we worry only about how the 6 sons are seated.

Answer 14

Okay...

Answer 15

That works out to be 720 different seating arrangements.

Answer 16

Okay well thank you.

Answer 17

@okdutchman7 look at the link

http://mathworld.wolfram.com/CircularPermutation.html

at the examples of arranging 3 people and arranging 4 people

There are 2 ways to arrange 3 people around a circle since (3-1)! = 2! = 2
There are 6 ways to arrange 4 people around a circle because (4-1)! = 3! = 6

The exclamation mark means factorial

http://www.purplemath.com/modules/factorial.htm