Question

in how many ways can 6 people be arranged in a circle if 2 particular people are always separated?

Answer 1

how many different places can the two people take in the circle of six of they cannot sit next to each other?

Answer 2

if they*

Answer 3

yes

Answer 4

question above?here is our circle of six.
imagine the two people who cannot sit next to each other.
How many different configurations can they be in?

Answer 5

say joe sits at 1
and bob can't sit next to joe, where can he sit?it looks like there are three possible places bob can sit for every place joe might sit. what about the rest? how many permutations can they take on for each possible configuration of joe and bob?

Answer 6

errh is it 3?

Answer 7

does the order matter?
does it matter who sits where?
or is it just the relation between the people?

Answer 8

i dont think order matters

Answer 9

Maybe you could calculate all the possible combinations and just substract the one in which the 2 people are next each other

Answer 10

then it is three
what about the rest of the people, how many configurations can they take on for each configuration of bob and joe?

Answer 11

how many seats are there beside bob and joe?

Answer 12

4

Answer 13

and how many permutations can those four seats have for each configuration of bob and joe?

Answer 14

ohhhhhhhhhhhhhhhhh
i think i have got it, is it 4! x 3=72?

Answer 15

should be if the order doesn't matter :)

Answer 16

YAYA! THANK YOU

Answer 17

one more thing could you help me check this question because my ans is different to the one in the solution book

Answer 18

which one?
new?

Answer 19

in how many ways can 5 boys and 3 women be arranged in a row?

Answer 20

i don't think there is any order so would it be 8!

Answer 21

do you know 8! is wrong?
because that's what it looks like to me too.

Answer 22

oh ok well the solution books says the answ is 4320, maybe you could work from this

Answer 23

it looks like
8!=40320
with a zero missing to me :/
I'd have to think of another solution...

Answer 24

haha that is exactly what i thought

Answer 25

I hope you don't mind me going back to the first question for a moment. The answer of 72, is that the final answer or is that just the amount for the first person in one position? Wouldn't the answer be 72 * 6 to account for each position the first of the two people could be in?

I'm going to be taking combinatorics in the upcoming semester, and I am guessing that I will be seeing alot of these kind of problems.

btw, I agree with 8! on the second problem as well, but can't say for sure that is right.

Answer 26

The first one had 2 people who could take the drivers seat, so that's just one seat to be filled by the people, so you don't have to look for other positions of the people.
When each of them is driving the rest of the people have 5! permutations of seats in the car. Since there are only two possible configurations for the driver, and 5! permutations of the remaining seats for each driver, the total is
2(5!)=240
I really don't think I know about this last one though...

Answer 27

I'd imagine across and Zarkon do, but they are being quiet...

Answer 28

oh you meant the first question on this post ChrisS?

Answer 29

Oh, yes it would be 72*6 if where they sat mattered, but we decided it didn't, just the relative configurations of the people.

Answer 30

I meant the first problem in this specific thread, the circle with six positions and two specific people cant sit next to each other. so for a given position of the first person, there are three spots the second person can be. Add in the other four positions where the order doesn't matter and you get the answer you arrived at of 3(4!)=72. But if you shift the first person through each of the 6 positions, doesn't each of those have to be added in to the total for a total of 72*6 or 432?

Answer 31

ok... I see you answered while I was typing lol

Answer 32

since there are only 3 configurations of the two people if it doesn't matter where they sat. If it does you get your answer
yep, sorry for the miscommunication

Answer 33

Well I think the four extra people it doesn't matter, but since it does matter between the two specific people, then it seems like with respect to the first person that each scenario should be accounted for.

Answer 34

That is the other possibility, so we should point that out to the questioner:
@virtus: If the seating matters then the answer is what ChrisS said, if not then it is what we found earlier.
still don't know about your last one.

Answer 35

AWWW THANK YOU SO MUCH GUYS, but i think the question does not take order into mind as in the solution book the answer is 72

Answer 36

there you go...
happy to help!

Answer 37

Yeah I was sitting here thinking about it and I guess it all boils down to "spots" being occupied on a circle which is how I was imagining it, or just a circle with 6 people in various configurations which by the answer is obviously how it was intended.