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

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

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question above?|dw:1325222124878:dw|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?
say joe sits at 1 and bob can't sit next to joe, where can he sit?|dw:1325222254760:dw|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?
errh is it 3?
does the order matter? does it matter who sits where? or is it just the relation between the people?
i dont think order matters
Maybe you could calculate all the possible combinations and just substract the one in which the 2 people are next each other
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?
how many seats are there beside bob and joe?
and how many permutations can those four seats have for each configuration of bob and joe?
ohhhhhhhhhhhhhhhhh i think i have got it, is it 4! x 3=72?
should be if the order doesn't matter :)
one more thing could you help me check this question because my ans is different to the one in the solution book
which one? new?
in how many ways can 5 boys and 3 women be arranged in a row?
i don't think there is any order so would it be 8!
do you know 8! is wrong? because that's what it looks like to me too.
oh ok well the solution books says the answ is 4320, maybe you could work from this
it looks like 8!=40320 with a zero missing to me :/ I'd have to think of another solution...
haha that is exactly what i thought
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.
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...
I'd imagine across and Zarkon do, but they are being quiet...
oh you meant the first question on this post ChrisS?
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.
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?
ok... I see you answered while I was typing lol
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
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.
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.
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
there you go... happy to help!
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.
however, i tried your method turning test on another similar question and i don't think i'm quite getting the right answer
father, mother and 6 children stand in a ring. In how many ways can they be arranged if father and mother are not to stand together
so if we fix the father then the mother has 4 alternative places where she can stand right? and the 6 kids can stand in any order so they will be 6!
I got 5 alternative places when I drew out my circle of 8 spots. ^ for the kids and two for the parents.
*6 for the kids
ohhhhhhh indeed my bad
sorrry that was my mistake hehehehe
it happens :)
ok heres another question i'm stuck on How many even numbers of 4 digits can be formed with the figures 3,4,7,8 if (a) no figure is repeated, (b) if repetitions are allowed

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