How many different arrangements are there if Bob and Sally must always be seated next to each other?












0












$begingroup$


How many different arrangements are there in which Bob, Sally and $n$ other people sit down in a row of $n+3$ chairs if Bob and Sally must always be seated next to each other?



I tried putting Bob and Sally next to each other in the first two chairs so then there are $2 times n!$ arrangements but then I need to move them to the 2nd and 3rd chair and so on. Not exactly sure how to do this.






combinatorics permutations







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    What have you tried? Where are you stuck?
    $endgroup$
    – Arthur
    Dec 7 '18 at 9:43










  • $begingroup$
    I tired putting bob and sally next to each other in the first two chairs so then there is 2x(n!) arrangements but then I need to move them to the 2nd and 3rd chair and so on. Not exactly sure how to do this
    $endgroup$
    – user607735
    Dec 7 '18 at 9:45






  • 1




    $begingroup$
    Why not try to get an answer for $n = 0, 1$ or $2$ first to get a feel for what's going on?
    $endgroup$
    – Arthur
    Dec 7 '18 at 9:47










  • $begingroup$
    Is the problem stated correctly, so one seat remains empty?
    $endgroup$
    – Christoph
    Dec 7 '18 at 9:53










  • $begingroup$
    Please read this tutorial on how to typeset mathematics on this site.
    $endgroup$
    – N. F. Taussig
    Dec 7 '18 at 10:07
















0












$begingroup$


How many different arrangements are there in which Bob, Sally and $n$ other people sit down in a row of $n+3$ chairs if Bob and Sally must always be seated next to each other?



I tried putting Bob and Sally next to each other in the first two chairs so then there are $2 times n!$ arrangements but then I need to move them to the 2nd and 3rd chair and so on. Not exactly sure how to do this.






combinatorics permutations







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    What have you tried? Where are you stuck?
    $endgroup$
    – Arthur
    Dec 7 '18 at 9:43










  • $begingroup$
    I tired putting bob and sally next to each other in the first two chairs so then there is 2x(n!) arrangements but then I need to move them to the 2nd and 3rd chair and so on. Not exactly sure how to do this
    $endgroup$
    – user607735
    Dec 7 '18 at 9:45






  • 1




    $begingroup$
    Why not try to get an answer for $n = 0, 1$ or $2$ first to get a feel for what's going on?
    $endgroup$
    – Arthur
    Dec 7 '18 at 9:47










  • $begingroup$
    Is the problem stated correctly, so one seat remains empty?
    $endgroup$
    – Christoph
    Dec 7 '18 at 9:53










  • $begingroup$
    Please read this tutorial on how to typeset mathematics on this site.
    $endgroup$
    – N. F. Taussig
    Dec 7 '18 at 10:07














0












0








0





$begingroup$


How many different arrangements are there in which Bob, Sally and $n$ other people sit down in a row of $n+3$ chairs if Bob and Sally must always be seated next to each other?



I tried putting Bob and Sally next to each other in the first two chairs so then there are $2 times n!$ arrangements but then I need to move them to the 2nd and 3rd chair and so on. Not exactly sure how to do this.






combinatorics permutations







share|cite|improve this question











$endgroup$




How many different arrangements are there in which Bob, Sally and $n$ other people sit down in a row of $n+3$ chairs if Bob and Sally must always be seated next to each other?



I tried putting Bob and Sally next to each other in the first two chairs so then there are $2 times n!$ arrangements but then I need to move them to the 2nd and 3rd chair and so on. Not exactly sure how to do this.






combinatorics permutations




combinatorics permutations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 7 '18 at 10:07









N. F. Taussig

44.1k93356




44.1k93356










asked Dec 7 '18 at 9:39









user607735user607735

103




103








  • 3




    $begingroup$
    What have you tried? Where are you stuck?
    $endgroup$
    – Arthur
    Dec 7 '18 at 9:43










  • $begingroup$
    I tired putting bob and sally next to each other in the first two chairs so then there is 2x(n!) arrangements but then I need to move them to the 2nd and 3rd chair and so on. Not exactly sure how to do this
    $endgroup$
    – user607735
    Dec 7 '18 at 9:45






  • 1




    $begingroup$
    Why not try to get an answer for $n = 0, 1$ or $2$ first to get a feel for what's going on?
    $endgroup$
    – Arthur
    Dec 7 '18 at 9:47










  • $begingroup$
    Is the problem stated correctly, so one seat remains empty?
    $endgroup$
    – Christoph
    Dec 7 '18 at 9:53










  • $begingroup$
    Please read this tutorial on how to typeset mathematics on this site.
    $endgroup$
    – N. F. Taussig
    Dec 7 '18 at 10:07














  • 3




    $begingroup$
    What have you tried? Where are you stuck?
    $endgroup$
    – Arthur
    Dec 7 '18 at 9:43










  • $begingroup$
    I tired putting bob and sally next to each other in the first two chairs so then there is 2x(n!) arrangements but then I need to move them to the 2nd and 3rd chair and so on. Not exactly sure how to do this
    $endgroup$
    – user607735
    Dec 7 '18 at 9:45






  • 1




    $begingroup$
    Why not try to get an answer for $n = 0, 1$ or $2$ first to get a feel for what's going on?
    $endgroup$
    – Arthur
    Dec 7 '18 at 9:47










  • $begingroup$
    Is the problem stated correctly, so one seat remains empty?
    $endgroup$
    – Christoph
    Dec 7 '18 at 9:53










  • $begingroup$
    Please read this tutorial on how to typeset mathematics on this site.
    $endgroup$
    – N. F. Taussig
    Dec 7 '18 at 10:07








3




3




$begingroup$
What have you tried? Where are you stuck?
$endgroup$
– Arthur
Dec 7 '18 at 9:43




$begingroup$
What have you tried? Where are you stuck?
$endgroup$
– Arthur
Dec 7 '18 at 9:43












$begingroup$
I tired putting bob and sally next to each other in the first two chairs so then there is 2x(n!) arrangements but then I need to move them to the 2nd and 3rd chair and so on. Not exactly sure how to do this
$endgroup$
– user607735
Dec 7 '18 at 9:45




$begingroup$
I tired putting bob and sally next to each other in the first two chairs so then there is 2x(n!) arrangements but then I need to move them to the 2nd and 3rd chair and so on. Not exactly sure how to do this
$endgroup$
– user607735
Dec 7 '18 at 9:45




1




1




$begingroup$
Why not try to get an answer for $n = 0, 1$ or $2$ first to get a feel for what's going on?
$endgroup$
– Arthur
Dec 7 '18 at 9:47




$begingroup$
Why not try to get an answer for $n = 0, 1$ or $2$ first to get a feel for what's going on?
$endgroup$
– Arthur
Dec 7 '18 at 9:47












$begingroup$
Is the problem stated correctly, so one seat remains empty?
$endgroup$
– Christoph
Dec 7 '18 at 9:53




$begingroup$
Is the problem stated correctly, so one seat remains empty?
$endgroup$
– Christoph
Dec 7 '18 at 9:53












$begingroup$
Please read this tutorial on how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 7 '18 at 10:07




$begingroup$
Please read this tutorial on how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 7 '18 at 10:07










3 Answers
3






active

oldest

votes


















2












$begingroup$

Hint: Consider Bob and Sally one unit taking two seats. So you have to place $n+1$ units at $n+2$ spots. In the end multiply by $2$ to account for Bob sitting on the left or right of Sally.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    yes so it could be writing as n+2 C n+1
    $endgroup$
    – user607735
    Dec 7 '18 at 9:55






  • 1




    $begingroup$
    Writing what as $binom{n+2}{n+1}$? The number of ways to seat the people is something else.
    $endgroup$
    – Christoph
    Dec 7 '18 at 9:56



















0












$begingroup$

Treat Bob and Sally as the same person (say, V) that takes up $2$ spaces, i.e. you put it in chair $m$ and then you can't put anyone in chair $m+1$. This means V cannot be in chair $n+3$. So, we place $V$ first in $n+2$ seats, and since it takes up $2$ seats, we now have $n+1$ seats for $n$ people. This means that the number of permutations is $(n+2)!$. Now, we can have $2$ "states" for $V$: one where Bob sits on chair $m$ and Sally on chair $m+1$, or Sally sits on chair $m$ and Bob sits on $m+1$. So, we multiply by $2$ to account, for a grand total of $2(n+2)!$ seatings arrangements.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    You can evaluate the arrangements separately.



    So for instance, the different arrangements in which Bob and Sally sit together are $$(n+2)times 2$$
    Note that deciding whether Bob or Sally sits to the right doubles the possibilities.



    Now, all possible arrangements in which $n$ different people sit (it doesn't matter whether together or not) is simply $$n!$$



    All possibilities are hence $$n!times (n+2)times2$$
    Note, however, that you also have to consider the gap; the total amount of possibilities is thus $$n!times (n+2)times 2times(n+1)=2times(n+2)!$$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      You could also just exchange the gap by an additional person (since there is only one) and obtain $(n+1)!$ instead of $n!$ immediately.
      $endgroup$
      – Christoph
      Dec 7 '18 at 10:00












    • $begingroup$
      @Christoph you're definitely right!
      $endgroup$
      – Dr. Mathva
      Dec 7 '18 at 10:26











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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    Hint: Consider Bob and Sally one unit taking two seats. So you have to place $n+1$ units at $n+2$ spots. In the end multiply by $2$ to account for Bob sitting on the left or right of Sally.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      yes so it could be writing as n+2 C n+1
      $endgroup$
      – user607735
      Dec 7 '18 at 9:55






    • 1




      $begingroup$
      Writing what as $binom{n+2}{n+1}$? The number of ways to seat the people is something else.
      $endgroup$
      – Christoph
      Dec 7 '18 at 9:56
















    2












    $begingroup$

    Hint: Consider Bob and Sally one unit taking two seats. So you have to place $n+1$ units at $n+2$ spots. In the end multiply by $2$ to account for Bob sitting on the left or right of Sally.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      yes so it could be writing as n+2 C n+1
      $endgroup$
      – user607735
      Dec 7 '18 at 9:55






    • 1




      $begingroup$
      Writing what as $binom{n+2}{n+1}$? The number of ways to seat the people is something else.
      $endgroup$
      – Christoph
      Dec 7 '18 at 9:56














    2












    2








    2





    $begingroup$

    Hint: Consider Bob and Sally one unit taking two seats. So you have to place $n+1$ units at $n+2$ spots. In the end multiply by $2$ to account for Bob sitting on the left or right of Sally.






    share|cite|improve this answer











    $endgroup$



    Hint: Consider Bob and Sally one unit taking two seats. So you have to place $n+1$ units at $n+2$ spots. In the end multiply by $2$ to account for Bob sitting on the left or right of Sally.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Dec 7 '18 at 10:01

























    answered Dec 7 '18 at 9:52









    ChristophChristoph

    12k1642




    12k1642












    • $begingroup$
      yes so it could be writing as n+2 C n+1
      $endgroup$
      – user607735
      Dec 7 '18 at 9:55






    • 1




      $begingroup$
      Writing what as $binom{n+2}{n+1}$? The number of ways to seat the people is something else.
      $endgroup$
      – Christoph
      Dec 7 '18 at 9:56


















    • $begingroup$
      yes so it could be writing as n+2 C n+1
      $endgroup$
      – user607735
      Dec 7 '18 at 9:55






    • 1




      $begingroup$
      Writing what as $binom{n+2}{n+1}$? The number of ways to seat the people is something else.
      $endgroup$
      – Christoph
      Dec 7 '18 at 9:56
















    $begingroup$
    yes so it could be writing as n+2 C n+1
    $endgroup$
    – user607735
    Dec 7 '18 at 9:55




    $begingroup$
    yes so it could be writing as n+2 C n+1
    $endgroup$
    – user607735
    Dec 7 '18 at 9:55




    1




    1




    $begingroup$
    Writing what as $binom{n+2}{n+1}$? The number of ways to seat the people is something else.
    $endgroup$
    – Christoph
    Dec 7 '18 at 9:56




    $begingroup$
    Writing what as $binom{n+2}{n+1}$? The number of ways to seat the people is something else.
    $endgroup$
    – Christoph
    Dec 7 '18 at 9:56











    0












    $begingroup$

    Treat Bob and Sally as the same person (say, V) that takes up $2$ spaces, i.e. you put it in chair $m$ and then you can't put anyone in chair $m+1$. This means V cannot be in chair $n+3$. So, we place $V$ first in $n+2$ seats, and since it takes up $2$ seats, we now have $n+1$ seats for $n$ people. This means that the number of permutations is $(n+2)!$. Now, we can have $2$ "states" for $V$: one where Bob sits on chair $m$ and Sally on chair $m+1$, or Sally sits on chair $m$ and Bob sits on $m+1$. So, we multiply by $2$ to account, for a grand total of $2(n+2)!$ seatings arrangements.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Treat Bob and Sally as the same person (say, V) that takes up $2$ spaces, i.e. you put it in chair $m$ and then you can't put anyone in chair $m+1$. This means V cannot be in chair $n+3$. So, we place $V$ first in $n+2$ seats, and since it takes up $2$ seats, we now have $n+1$ seats for $n$ people. This means that the number of permutations is $(n+2)!$. Now, we can have $2$ "states" for $V$: one where Bob sits on chair $m$ and Sally on chair $m+1$, or Sally sits on chair $m$ and Bob sits on $m+1$. So, we multiply by $2$ to account, for a grand total of $2(n+2)!$ seatings arrangements.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Treat Bob and Sally as the same person (say, V) that takes up $2$ spaces, i.e. you put it in chair $m$ and then you can't put anyone in chair $m+1$. This means V cannot be in chair $n+3$. So, we place $V$ first in $n+2$ seats, and since it takes up $2$ seats, we now have $n+1$ seats for $n$ people. This means that the number of permutations is $(n+2)!$. Now, we can have $2$ "states" for $V$: one where Bob sits on chair $m$ and Sally on chair $m+1$, or Sally sits on chair $m$ and Bob sits on $m+1$. So, we multiply by $2$ to account, for a grand total of $2(n+2)!$ seatings arrangements.






        share|cite|improve this answer









        $endgroup$



        Treat Bob and Sally as the same person (say, V) that takes up $2$ spaces, i.e. you put it in chair $m$ and then you can't put anyone in chair $m+1$. This means V cannot be in chair $n+3$. So, we place $V$ first in $n+2$ seats, and since it takes up $2$ seats, we now have $n+1$ seats for $n$ people. This means that the number of permutations is $(n+2)!$. Now, we can have $2$ "states" for $V$: one where Bob sits on chair $m$ and Sally on chair $m+1$, or Sally sits on chair $m$ and Bob sits on $m+1$. So, we multiply by $2$ to account, for a grand total of $2(n+2)!$ seatings arrangements.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 7 '18 at 9:56









        KykyKyky

        460213




        460213























            0












            $begingroup$

            You can evaluate the arrangements separately.



            So for instance, the different arrangements in which Bob and Sally sit together are $$(n+2)times 2$$
            Note that deciding whether Bob or Sally sits to the right doubles the possibilities.



            Now, all possible arrangements in which $n$ different people sit (it doesn't matter whether together or not) is simply $$n!$$



            All possibilities are hence $$n!times (n+2)times2$$
            Note, however, that you also have to consider the gap; the total amount of possibilities is thus $$n!times (n+2)times 2times(n+1)=2times(n+2)!$$






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              You could also just exchange the gap by an additional person (since there is only one) and obtain $(n+1)!$ instead of $n!$ immediately.
              $endgroup$
              – Christoph
              Dec 7 '18 at 10:00












            • $begingroup$
              @Christoph you're definitely right!
              $endgroup$
              – Dr. Mathva
              Dec 7 '18 at 10:26
















            0












            $begingroup$

            You can evaluate the arrangements separately.



            So for instance, the different arrangements in which Bob and Sally sit together are $$(n+2)times 2$$
            Note that deciding whether Bob or Sally sits to the right doubles the possibilities.



            Now, all possible arrangements in which $n$ different people sit (it doesn't matter whether together or not) is simply $$n!$$



            All possibilities are hence $$n!times (n+2)times2$$
            Note, however, that you also have to consider the gap; the total amount of possibilities is thus $$n!times (n+2)times 2times(n+1)=2times(n+2)!$$






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              You could also just exchange the gap by an additional person (since there is only one) and obtain $(n+1)!$ instead of $n!$ immediately.
              $endgroup$
              – Christoph
              Dec 7 '18 at 10:00












            • $begingroup$
              @Christoph you're definitely right!
              $endgroup$
              – Dr. Mathva
              Dec 7 '18 at 10:26














            0












            0








            0





            $begingroup$

            You can evaluate the arrangements separately.



            So for instance, the different arrangements in which Bob and Sally sit together are $$(n+2)times 2$$
            Note that deciding whether Bob or Sally sits to the right doubles the possibilities.



            Now, all possible arrangements in which $n$ different people sit (it doesn't matter whether together or not) is simply $$n!$$



            All possibilities are hence $$n!times (n+2)times2$$
            Note, however, that you also have to consider the gap; the total amount of possibilities is thus $$n!times (n+2)times 2times(n+1)=2times(n+2)!$$






            share|cite|improve this answer









            $endgroup$



            You can evaluate the arrangements separately.



            So for instance, the different arrangements in which Bob and Sally sit together are $$(n+2)times 2$$
            Note that deciding whether Bob or Sally sits to the right doubles the possibilities.



            Now, all possible arrangements in which $n$ different people sit (it doesn't matter whether together or not) is simply $$n!$$



            All possibilities are hence $$n!times (n+2)times2$$
            Note, however, that you also have to consider the gap; the total amount of possibilities is thus $$n!times (n+2)times 2times(n+1)=2times(n+2)!$$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 7 '18 at 9:58









            Dr. MathvaDr. Mathva

            1,102317




            1,102317












            • $begingroup$
              You could also just exchange the gap by an additional person (since there is only one) and obtain $(n+1)!$ instead of $n!$ immediately.
              $endgroup$
              – Christoph
              Dec 7 '18 at 10:00












            • $begingroup$
              @Christoph you're definitely right!
              $endgroup$
              – Dr. Mathva
              Dec 7 '18 at 10:26


















            • $begingroup$
              You could also just exchange the gap by an additional person (since there is only one) and obtain $(n+1)!$ instead of $n!$ immediately.
              $endgroup$
              – Christoph
              Dec 7 '18 at 10:00












            • $begingroup$
              @Christoph you're definitely right!
              $endgroup$
              – Dr. Mathva
              Dec 7 '18 at 10:26
















            $begingroup$
            You could also just exchange the gap by an additional person (since there is only one) and obtain $(n+1)!$ instead of $n!$ immediately.
            $endgroup$
            – Christoph
            Dec 7 '18 at 10:00






            $begingroup$
            You could also just exchange the gap by an additional person (since there is only one) and obtain $(n+1)!$ instead of $n!$ immediately.
            $endgroup$
            – Christoph
            Dec 7 '18 at 10:00














            $begingroup$
            @Christoph you're definitely right!
            $endgroup$
            – Dr. Mathva
            Dec 7 '18 at 10:26




            $begingroup$
            @Christoph you're definitely right!
            $endgroup$
            – Dr. Mathva
            Dec 7 '18 at 10:26


















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            Bundesstraße 106

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            Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück
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            Le Mesnil-Réaume

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            Le Mesnil-Réaume Region Normandie Département Seine-Maritime Arrondissement Dieppe Kanton Eu Gemeindeverband Villes Sœurs Koordinaten 49° 58′  N , 1° 27′  O 49.9725 1.4463888888889 Koordinaten: 49° 58′  N , 1° 27′  O Höhe 92–156 m Fläche 5,55 km 2 Einwohner 793 (1. Januar 2016) Bevölkerungsdichte 143 Einw./km 2 Postleitzahl 76260 INSEE-Code 76435 Website http://www.mesnil-reaume.net/ Rathaus (Mairie) und Schule von Le Mesnil-Réaume Le Mesnil-Réaume ist eine französische Gemeinde mit 793 Einwohnern (Stand: 1. Januar 2016) im Département Seine-Maritime in der Region Normandie (vor 2016 Haute-Normandie). Sie gehört zum Arrondissement Dieppe und zum Kanton Eu. Die Einwohner werden Mesnillais genannt. Inhaltsverzeichnis 1 Geographie 2 Bevölkerungsentwicklung 3 Sehenswürdigkeiten 4 Weblinks Geographie | Le Mesnil-Réaume liegt etwa 43 Kilomet
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            Ida-Boy-Ed-Garten

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            Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich
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