How many ways to place identical boxes in a row?











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How many ways there are to arrange ten (or n) identical boxes in a row of ten (or n) places when stacking is allowed? I guess you need to find how many different arrays there are but how?






combinatorics







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  • 1




    If you consider the height of stack $j$ as $x_j$, then you're looking at the equation $$ x_1 + x_2 + x_3 + x_4 + ldots + x_{10} = n $$ The number of solutions can be found with the "stars and bars" method.
    – Matti P.
    Nov 15 at 11:21












  • This is an example of a composition problem.
    – N. F. Taussig
    Nov 15 at 17:13










  • Thanks a bunch. I was quite sure this was some classical problem but just couldn't figure it out myself
    – user2566933
    Nov 16 at 19:51















up vote
0
down vote

favorite












How many ways there are to arrange ten (or n) identical boxes in a row of ten (or n) places when stacking is allowed? I guess you need to find how many different arrays there are but how?






combinatorics







share|cite|improve this question


















  • 1




    If you consider the height of stack $j$ as $x_j$, then you're looking at the equation $$ x_1 + x_2 + x_3 + x_4 + ldots + x_{10} = n $$ The number of solutions can be found with the "stars and bars" method.
    – Matti P.
    Nov 15 at 11:21












  • This is an example of a composition problem.
    – N. F. Taussig
    Nov 15 at 17:13










  • Thanks a bunch. I was quite sure this was some classical problem but just couldn't figure it out myself
    – user2566933
    Nov 16 at 19:51













up vote
0
down vote

favorite









up vote
0
down vote

favorite











How many ways there are to arrange ten (or n) identical boxes in a row of ten (or n) places when stacking is allowed? I guess you need to find how many different arrays there are but how?






combinatorics







share|cite|improve this question













How many ways there are to arrange ten (or n) identical boxes in a row of ten (or n) places when stacking is allowed? I guess you need to find how many different arrays there are but how?






combinatorics




combinatorics






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 15 at 11:17









user2566933

62




62








  • 1




    If you consider the height of stack $j$ as $x_j$, then you're looking at the equation $$ x_1 + x_2 + x_3 + x_4 + ldots + x_{10} = n $$ The number of solutions can be found with the "stars and bars" method.
    – Matti P.
    Nov 15 at 11:21












  • This is an example of a composition problem.
    – N. F. Taussig
    Nov 15 at 17:13










  • Thanks a bunch. I was quite sure this was some classical problem but just couldn't figure it out myself
    – user2566933
    Nov 16 at 19:51














  • 1




    If you consider the height of stack $j$ as $x_j$, then you're looking at the equation $$ x_1 + x_2 + x_3 + x_4 + ldots + x_{10} = n $$ The number of solutions can be found with the "stars and bars" method.
    – Matti P.
    Nov 15 at 11:21












  • This is an example of a composition problem.
    – N. F. Taussig
    Nov 15 at 17:13










  • Thanks a bunch. I was quite sure this was some classical problem but just couldn't figure it out myself
    – user2566933
    Nov 16 at 19:51








1




1




If you consider the height of stack $j$ as $x_j$, then you're looking at the equation $$ x_1 + x_2 + x_3 + x_4 + ldots + x_{10} = n $$ The number of solutions can be found with the "stars and bars" method.
– Matti P.
Nov 15 at 11:21






If you consider the height of stack $j$ as $x_j$, then you're looking at the equation $$ x_1 + x_2 + x_3 + x_4 + ldots + x_{10} = n $$ The number of solutions can be found with the "stars and bars" method.
– Matti P.
Nov 15 at 11:21














This is an example of a composition problem.
– N. F. Taussig
Nov 15 at 17:13




This is an example of a composition problem.
– N. F. Taussig
Nov 15 at 17:13












Thanks a bunch. I was quite sure this was some classical problem but just couldn't figure it out myself
– user2566933
Nov 16 at 19:51




Thanks a bunch. I was quite sure this was some classical problem but just couldn't figure it out myself
– user2566933
Nov 16 at 19:51










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Arrange the $n$ boxes in a row, like so:
$$square square square square square square square square square square square square ldots square square square square tag{1}$$
There are $n-1$ spaces in between. "Stacking" the boxes means inserting a $|$ into some of the spaces in $(1)$, indicating when a new stack shall begin. Since there are $2^{n-1}$ ways to choose a subset of the $n-1$ spaces there are $2^{n-1}$ ways to arrange the boxes in a row of nonempty stacks.






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    Arrange the $n$ boxes in a row, like so:
    $$square square square square square square square square square square square square ldots square square square square tag{1}$$
    There are $n-1$ spaces in between. "Stacking" the boxes means inserting a $|$ into some of the spaces in $(1)$, indicating when a new stack shall begin. Since there are $2^{n-1}$ ways to choose a subset of the $n-1$ spaces there are $2^{n-1}$ ways to arrange the boxes in a row of nonempty stacks.






    share|cite|improve this answer

























      up vote
      2
      down vote













      Arrange the $n$ boxes in a row, like so:
      $$square square square square square square square square square square square square ldots square square square square tag{1}$$
      There are $n-1$ spaces in between. "Stacking" the boxes means inserting a $|$ into some of the spaces in $(1)$, indicating when a new stack shall begin. Since there are $2^{n-1}$ ways to choose a subset of the $n-1$ spaces there are $2^{n-1}$ ways to arrange the boxes in a row of nonempty stacks.






      share|cite|improve this answer























        up vote
        2
        down vote










        up vote
        2
        down vote









        Arrange the $n$ boxes in a row, like so:
        $$square square square square square square square square square square square square ldots square square square square tag{1}$$
        There are $n-1$ spaces in between. "Stacking" the boxes means inserting a $|$ into some of the spaces in $(1)$, indicating when a new stack shall begin. Since there are $2^{n-1}$ ways to choose a subset of the $n-1$ spaces there are $2^{n-1}$ ways to arrange the boxes in a row of nonempty stacks.






        share|cite|improve this answer












        Arrange the $n$ boxes in a row, like so:
        $$square square square square square square square square square square square square ldots square square square square tag{1}$$
        There are $n-1$ spaces in between. "Stacking" the boxes means inserting a $|$ into some of the spaces in $(1)$, indicating when a new stack shall begin. Since there are $2^{n-1}$ ways to choose a subset of the $n-1$ spaces there are $2^{n-1}$ ways to arrange the boxes in a row of nonempty stacks.







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        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 15 at 14:00









        Christian Blatter

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