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
asked Nov 15 at 11:17
user2566933
62
add a comment |
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
asked Nov 15 at 11:17
user2566933
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
add a comment |
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
asked Nov 15 at 11:17
user2566933
62
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
asked Nov 15 at 11:17
user2566933
62
asked Nov 15 at 11:17
user2566933
62
asked Nov 15 at 11:17
user2566933
62
asked Nov 15 at 11:17
user2566933
62
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
add a comment |
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
add a comment |
1 Answer
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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.
answered Nov 15 at 14:00
Christian Blatter
170k7111324
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
answered Nov 15 at 14:00
Christian Blatter
170k7111324
add a comment |
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.
answered Nov 15 at 14:00
Christian Blatter
170k7111324
add a comment |
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.
answered Nov 15 at 14:00
Christian Blatter
170k7111324
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.
answered Nov 15 at 14:00
Christian Blatter
170k7111324
answered Nov 15 at 14:00
Christian Blatter
170k7111324
answered Nov 15 at 14:00
Christian Blatter
170k7111324
answered Nov 15 at 14:00
Christian Blatter
170k7111324
170k7111324
add a comment |
add a comment |
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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