circular r-permutations of n












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My book, Discrete Mathematics And Its Applications by K.H.Rosen asks to find a formula for circular r-permutations of n people. That is, sitting of r of these people around a table when two sittings are considered equal if they look the same by rotation of the table.



As I searched, the answer must be, P(n,r)/2r but the book says something different:



The books answer.



So my problem is why? Why do we divide by r and not by 2 at the end?
And why didn't "design the head" step divide the answer by r?










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  • Think this way: How many ways can we seat $r$ people around a circular table? This is $r!/r$. To seat $r$ out of $n$ people, we first choose $r$ people in $binom{n}{r}$ ways and seat them in $(r-1)!$ ways.
    – Muralidharan
    Dec 4 '18 at 16:29










  • "As I searched, the answer must be, P(n,r)/2r" -- Since there are r seats and rotations don't count, you would divide by r. Why would you divide by 2?
    – dcromley
    Dec 4 '18 at 16:34










  • I think the issue is that you're counting reflections as being equivalent, but the book isn't.
    – user3482749
    Dec 4 '18 at 16:36










  • I divide by 2 because the as the book says "sittings are considered same, if they are obtained from each other by rotating the table."
    – AK 12
    Dec 4 '18 at 16:40






  • 1




    "... rotating the table". That's where the r factor comes in. 2 is for reflection symmetry.
    – karakfa
    Dec 4 '18 at 17:43
















0














My book, Discrete Mathematics And Its Applications by K.H.Rosen asks to find a formula for circular r-permutations of n people. That is, sitting of r of these people around a table when two sittings are considered equal if they look the same by rotation of the table.



As I searched, the answer must be, P(n,r)/2r but the book says something different:



The books answer.



So my problem is why? Why do we divide by r and not by 2 at the end?
And why didn't "design the head" step divide the answer by r?










share|cite|improve this question






















  • Think this way: How many ways can we seat $r$ people around a circular table? This is $r!/r$. To seat $r$ out of $n$ people, we first choose $r$ people in $binom{n}{r}$ ways and seat them in $(r-1)!$ ways.
    – Muralidharan
    Dec 4 '18 at 16:29










  • "As I searched, the answer must be, P(n,r)/2r" -- Since there are r seats and rotations don't count, you would divide by r. Why would you divide by 2?
    – dcromley
    Dec 4 '18 at 16:34










  • I think the issue is that you're counting reflections as being equivalent, but the book isn't.
    – user3482749
    Dec 4 '18 at 16:36










  • I divide by 2 because the as the book says "sittings are considered same, if they are obtained from each other by rotating the table."
    – AK 12
    Dec 4 '18 at 16:40






  • 1




    "... rotating the table". That's where the r factor comes in. 2 is for reflection symmetry.
    – karakfa
    Dec 4 '18 at 17:43














0












0








0







My book, Discrete Mathematics And Its Applications by K.H.Rosen asks to find a formula for circular r-permutations of n people. That is, sitting of r of these people around a table when two sittings are considered equal if they look the same by rotation of the table.



As I searched, the answer must be, P(n,r)/2r but the book says something different:



The books answer.



So my problem is why? Why do we divide by r and not by 2 at the end?
And why didn't "design the head" step divide the answer by r?










share|cite|improve this question













My book, Discrete Mathematics And Its Applications by K.H.Rosen asks to find a formula for circular r-permutations of n people. That is, sitting of r of these people around a table when two sittings are considered equal if they look the same by rotation of the table.



As I searched, the answer must be, P(n,r)/2r but the book says something different:



The books answer.



So my problem is why? Why do we divide by r and not by 2 at the end?
And why didn't "design the head" step divide the answer by r?







combinatorics permutations






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asked Dec 4 '18 at 16:23









AK 12

11




11












  • Think this way: How many ways can we seat $r$ people around a circular table? This is $r!/r$. To seat $r$ out of $n$ people, we first choose $r$ people in $binom{n}{r}$ ways and seat them in $(r-1)!$ ways.
    – Muralidharan
    Dec 4 '18 at 16:29










  • "As I searched, the answer must be, P(n,r)/2r" -- Since there are r seats and rotations don't count, you would divide by r. Why would you divide by 2?
    – dcromley
    Dec 4 '18 at 16:34










  • I think the issue is that you're counting reflections as being equivalent, but the book isn't.
    – user3482749
    Dec 4 '18 at 16:36










  • I divide by 2 because the as the book says "sittings are considered same, if they are obtained from each other by rotating the table."
    – AK 12
    Dec 4 '18 at 16:40






  • 1




    "... rotating the table". That's where the r factor comes in. 2 is for reflection symmetry.
    – karakfa
    Dec 4 '18 at 17:43


















  • Think this way: How many ways can we seat $r$ people around a circular table? This is $r!/r$. To seat $r$ out of $n$ people, we first choose $r$ people in $binom{n}{r}$ ways and seat them in $(r-1)!$ ways.
    – Muralidharan
    Dec 4 '18 at 16:29










  • "As I searched, the answer must be, P(n,r)/2r" -- Since there are r seats and rotations don't count, you would divide by r. Why would you divide by 2?
    – dcromley
    Dec 4 '18 at 16:34










  • I think the issue is that you're counting reflections as being equivalent, but the book isn't.
    – user3482749
    Dec 4 '18 at 16:36










  • I divide by 2 because the as the book says "sittings are considered same, if they are obtained from each other by rotating the table."
    – AK 12
    Dec 4 '18 at 16:40






  • 1




    "... rotating the table". That's where the r factor comes in. 2 is for reflection symmetry.
    – karakfa
    Dec 4 '18 at 17:43
















Think this way: How many ways can we seat $r$ people around a circular table? This is $r!/r$. To seat $r$ out of $n$ people, we first choose $r$ people in $binom{n}{r}$ ways and seat them in $(r-1)!$ ways.
– Muralidharan
Dec 4 '18 at 16:29




Think this way: How many ways can we seat $r$ people around a circular table? This is $r!/r$. To seat $r$ out of $n$ people, we first choose $r$ people in $binom{n}{r}$ ways and seat them in $(r-1)!$ ways.
– Muralidharan
Dec 4 '18 at 16:29












"As I searched, the answer must be, P(n,r)/2r" -- Since there are r seats and rotations don't count, you would divide by r. Why would you divide by 2?
– dcromley
Dec 4 '18 at 16:34




"As I searched, the answer must be, P(n,r)/2r" -- Since there are r seats and rotations don't count, you would divide by r. Why would you divide by 2?
– dcromley
Dec 4 '18 at 16:34












I think the issue is that you're counting reflections as being equivalent, but the book isn't.
– user3482749
Dec 4 '18 at 16:36




I think the issue is that you're counting reflections as being equivalent, but the book isn't.
– user3482749
Dec 4 '18 at 16:36












I divide by 2 because the as the book says "sittings are considered same, if they are obtained from each other by rotating the table."
– AK 12
Dec 4 '18 at 16:40




I divide by 2 because the as the book says "sittings are considered same, if they are obtained from each other by rotating the table."
– AK 12
Dec 4 '18 at 16:40




1




1




"... rotating the table". That's where the r factor comes in. 2 is for reflection symmetry.
– karakfa
Dec 4 '18 at 17:43




"... rotating the table". That's where the r factor comes in. 2 is for reflection symmetry.
– karakfa
Dec 4 '18 at 17:43










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What you probably got the answer is the "bracelet problem", what is being asked is the "necklace problem". The difference is the bracelet is assumed to be equivalent under reflection, whereas necklace is not. see wikipedia article here






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    What you probably got the answer is the "bracelet problem", what is being asked is the "necklace problem". The difference is the bracelet is assumed to be equivalent under reflection, whereas necklace is not. see wikipedia article here






    share|cite|improve this answer


























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      What you probably got the answer is the "bracelet problem", what is being asked is the "necklace problem". The difference is the bracelet is assumed to be equivalent under reflection, whereas necklace is not. see wikipedia article here






      share|cite|improve this answer
























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        What you probably got the answer is the "bracelet problem", what is being asked is the "necklace problem". The difference is the bracelet is assumed to be equivalent under reflection, whereas necklace is not. see wikipedia article here






        share|cite|improve this answer












        What you probably got the answer is the "bracelet problem", what is being asked is the "necklace problem". The difference is the bracelet is assumed to be equivalent under reflection, whereas necklace is not. see wikipedia article here







        share|cite|improve this answer












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        answered Dec 4 '18 at 19:10









        karakfa

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