Circular Arrangement Problem

[warped kraken]

pls help I'm trash at perms and combs

At a family gathering, 6 adults and 3 children are randomly seated around a circular table. What is the number of possible seating arrangements if none of the children are to sit together?

[tough pine]

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[high temple]

the arrangement must be AcAcAc if none of the children are to be together

(A adult c child)

are rotations and reflections counted as different?

[warped kraken]

nope

thats why the formula to find circular arrangements is (n-1)!

[high temple]

ok

fix one adult in place to deal with rotations

and remind me to divide by 2 for reflections later

now the other adults are arranged in 2! ways and the children in 3! ways

[high temple]

wait… this takes rotations as the same but reflections as different

[warped kraken]

but there r 6 adults

[warped kraken]

not sure what u mean by reflections

[high temple]

oh wait i fully misread the question

my bad

[warped kraken]

allg

[high temple]

ok restart

hmm

we can position an adult directly to the left of each child

this should satisfy the condition

now arrange these 6 “clumps” around the table

[high temple]

is ACB the same as ABC, taken clockwise

[warped kraken]

nope dont think so

[warped kraken]

could u maybe draw it up

[high temple]

ok

[high temple]

i’m on phone

[warped kraken]

maybe draw it on microsoft onenote and take a ss?

[high temple]

AAAAAA ccc
pair each child up: Ac Ac Ac A A A
arrange freely: Ac A A Ac Ac A

idk how this would be drawn

[warped kraken]

so by doing that, I fix one adult at a certain point, so the remaining adults would be arranged in 5! ways, and the remaining children would be arranged in 3! ways, so is 3! x 5! the solution?

[high temple]

you would need to factor in the “assignment” of the adults at the start

i believe that is 6*5*4

[high temple]

the children are arranged with the adults

there is no need for the 3!

the first step introduces a factor of 120 and the second a factor of 5!

[warped kraken]

why the 9 x 8 x 7

[high temple]

first child can be given one of 9 adults

second, one of 8

[warped kraken]

but theres 6 adults

whered u get 9 from

[high temple]

from my overexcited brain

i’m sorry

[warped kraken]

it's allg bud take ur time and when ur confident u can teach me

[high temple]

no i’m sure this time

[warped kraken]

wait so could u restart

also my wifi is a bit dodgy so excuse my late replies

[high temple]

that would probably be for the best lol

6A 3c (adults, children)
give each child an adult to its left (6*5*4 ways to do this)
A A A Ac Ac Ac
6 clumps formed, which must be kept together when ordering
place them around the table in 5! ways using the formula

[warped kraken]

so the final answer is 5! ?

[high temple]

i think it is 6*5*4*5!

[warped kraken]

thats the right answer

but i dont get the point of the 6 x 5 x 4 part when ur doing 5!

because doesn't 5! take into account the ways u can arrange the clumps together already?

[high temple]

but the 6*5*4 describes how the clumps are created

[warped kraken]

so hows this different to 6 distinct objects placed around a circular table

[high temple]

because you make the clumps in different ways

basically you don’t have 6 distinct objects, you have 6 boxes and a few more objects, which you first pack into boxes in a number of ways, then the boxes are arranged in a circle

[warped kraken]

right thanks mate

[high temple]

sorry it took so many tries haha