Number of solutions of inequality
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I have the following combinatorics problem which I have no clue how to solve.
What is the number of solutions of the inequality $$sumlimits_i = 1^4 x_i ge 0 , x_i in - 1,1 $$
I know how to solve these kind of questions when the sum is equal rather than greater or equal than, and it equals to a number other than 0, which is a simple stars and bars problem but this one leaves me pretty much clueless. I thought of somehow transforming the problem to an equivalent problem:
$$sumlimits_i = 1^4 k_i ge 2 ,k_i in 0,2 $$
but I'm still stuck...Any help will be much appreciated :)
combinatorics
asked Aug 18 at 0:00
Amiros89
93
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up vote
1
down vote
favorite
I have the following combinatorics problem which I have no clue how to solve.
What is the number of solutions of the inequality $$sumlimits_i = 1^4 x_i ge 0 , x_i in - 1,1 $$
I know how to solve these kind of questions when the sum is equal rather than greater or equal than, and it equals to a number other than 0, which is a simple stars and bars problem but this one leaves me pretty much clueless. I thought of somehow transforming the problem to an equivalent problem:
$$sumlimits_i = 1^4 k_i ge 2 ,k_i in 0,2 $$
but I'm still stuck...Any help will be much appreciated :)
combinatorics
asked Aug 18 at 0:00
Amiros89
93
In the second inequality, $k_iin0,1$.
– Alexander Burstein
Aug 19 at 7:25
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have the following combinatorics problem which I have no clue how to solve.
What is the number of solutions of the inequality $$sumlimits_i = 1^4 x_i ge 0 , x_i in - 1,1 $$
I know how to solve these kind of questions when the sum is equal rather than greater or equal than, and it equals to a number other than 0, which is a simple stars and bars problem but this one leaves me pretty much clueless. I thought of somehow transforming the problem to an equivalent problem:
$$sumlimits_i = 1^4 k_i ge 2 ,k_i in 0,2 $$
but I'm still stuck...Any help will be much appreciated :)
combinatorics
asked Aug 18 at 0:00
Amiros89
93
I have the following combinatorics problem which I have no clue how to solve.
What is the number of solutions of the inequality $$sumlimits_i = 1^4 x_i ge 0 , x_i in - 1,1 $$
I know how to solve these kind of questions when the sum is equal rather than greater or equal than, and it equals to a number other than 0, which is a simple stars and bars problem but this one leaves me pretty much clueless. I thought of somehow transforming the problem to an equivalent problem:
$$sumlimits_i = 1^4 k_i ge 2 ,k_i in 0,2 $$
but I'm still stuck...Any help will be much appreciated :)
combinatorics
asked Aug 18 at 0:00
Amiros89
93
asked Aug 18 at 0:00
Amiros89
93
asked Aug 18 at 0:00
Amiros89
93
asked Aug 18 at 0:00
Amiros89
93
93
In the second inequality, $k_iin0,1$.
– Alexander Burstein
Aug 19 at 7:25
add a comment |Â
In the second inequality, $k_iin0,1$.
– Alexander Burstein
Aug 19 at 7:25
In the second inequality, $k_iin0,1$.
– Alexander Burstein
Aug 19 at 7:25
In the second inequality, $k_iin0,1$.
– Alexander Burstein
Aug 19 at 7:25
add a comment |Â
2 Answers
2
active
oldest
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up vote
4
down vote
HINT: All four variables or three of them or at least two of them must be equal to $1$.
answered Aug 18 at 0:12
Bumblebee
9,45012449
Thanks. So basically if I got it right this is the right solution? $$left( matrix 4 cr 2 cr right) + left( matrix 4 cr 3 cr right) + left( matrix 4 cr 4 cr right) = 6 + 4 + 1 = 11$$
– Amiros89
Aug 18 at 8:59
add a comment |Â
up vote
0
down vote
I know how to solve these kind of questions when the sum is equal rather than greater or equal than, and it equals to a number other than 0, which is a simple stars and bars problem
A lot of mathematics is about recognising how to transform something you don't know how to solve into something you do know how to solve. If you know how to solve $$sum_i=1^4 x_i = k, x_i in -1,1$$ then can you substitute the solution into $$sum_k ge 0 sum_i=1^4 x_i = k, x_i in -1,1$$ to solve the original problem?
(Note: $k=0$ isn't really a special case after all).
answered Aug 20 at 10:02
Peter Taylor
7,82012239
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
HINT: All four variables or three of them or at least two of them must be equal to $1$.
answered Aug 18 at 0:12
Bumblebee
9,45012449
Thanks. So basically if I got it right this is the right solution? $$left( matrix 4 cr 2 cr right) + left( matrix 4 cr 3 cr right) + left( matrix 4 cr 4 cr right) = 6 + 4 + 1 = 11$$
– Amiros89
Aug 18 at 8:59
add a comment |Â
up vote
4
down vote
HINT: All four variables or three of them or at least two of them must be equal to $1$.
answered Aug 18 at 0:12
Bumblebee
9,45012449
Thanks. So basically if I got it right this is the right solution? $$left( matrix 4 cr 2 cr right) + left( matrix 4 cr 3 cr right) + left( matrix 4 cr 4 cr right) = 6 + 4 + 1 = 11$$
– Amiros89
Aug 18 at 8:59
add a comment |Â
up vote
4
down vote
up vote
4
down vote
HINT: All four variables or three of them or at least two of them must be equal to $1$.
answered Aug 18 at 0:12
Bumblebee
9,45012449
HINT: All four variables or three of them or at least two of them must be equal to $1$.
answered Aug 18 at 0:12
Bumblebee
9,45012449
answered Aug 18 at 0:12
Bumblebee
9,45012449
answered Aug 18 at 0:12
Bumblebee
9,45012449
answered Aug 18 at 0:12
Bumblebee
9,45012449
9,45012449
Thanks. So basically if I got it right this is the right solution? $$left( matrix 4 cr 2 cr right) + left( matrix 4 cr 3 cr right) + left( matrix 4 cr 4 cr right) = 6 + 4 + 1 = 11$$
– Amiros89
Aug 18 at 8:59
add a comment |Â
Thanks. So basically if I got it right this is the right solution? $$left( matrix 4 cr 2 cr right) + left( matrix 4 cr 3 cr right) + left( matrix 4 cr 4 cr right) = 6 + 4 + 1 = 11$$
– Amiros89
Aug 18 at 8:59
Thanks. So basically if I got it right this is the right solution? $$left( matrix 4 cr 2 cr right) + left( matrix 4 cr 3 cr right) + left( matrix 4 cr 4 cr right) = 6 + 4 + 1 = 11$$
– Amiros89
Aug 18 at 8:59
Thanks. So basically if I got it right this is the right solution? $$left( matrix 4 cr 2 cr right) + left( matrix 4 cr 3 cr right) + left( matrix 4 cr 4 cr right) = 6 + 4 + 1 = 11$$
– Amiros89
Aug 18 at 8:59
add a comment |Â
up vote
0
down vote
I know how to solve these kind of questions when the sum is equal rather than greater or equal than, and it equals to a number other than 0, which is a simple stars and bars problem
A lot of mathematics is about recognising how to transform something you don't know how to solve into something you do know how to solve. If you know how to solve $$sum_i=1^4 x_i = k, x_i in -1,1$$ then can you substitute the solution into $$sum_k ge 0 sum_i=1^4 x_i = k, x_i in -1,1$$ to solve the original problem?
(Note: $k=0$ isn't really a special case after all).
answered Aug 20 at 10:02
Peter Taylor
7,82012239
add a comment |Â
up vote
0
down vote
I know how to solve these kind of questions when the sum is equal rather than greater or equal than, and it equals to a number other than 0, which is a simple stars and bars problem
A lot of mathematics is about recognising how to transform something you don't know how to solve into something you do know how to solve. If you know how to solve $$sum_i=1^4 x_i = k, x_i in -1,1$$ then can you substitute the solution into $$sum_k ge 0 sum_i=1^4 x_i = k, x_i in -1,1$$ to solve the original problem?
(Note: $k=0$ isn't really a special case after all).
answered Aug 20 at 10:02
Peter Taylor
7,82012239
add a comment |Â
up vote
0
down vote
up vote
0
down vote
I know how to solve these kind of questions when the sum is equal rather than greater or equal than, and it equals to a number other than 0, which is a simple stars and bars problem
A lot of mathematics is about recognising how to transform something you don't know how to solve into something you do know how to solve. If you know how to solve $$sum_i=1^4 x_i = k, x_i in -1,1$$ then can you substitute the solution into $$sum_k ge 0 sum_i=1^4 x_i = k, x_i in -1,1$$ to solve the original problem?
(Note: $k=0$ isn't really a special case after all).
answered Aug 20 at 10:02
Peter Taylor
7,82012239
I know how to solve these kind of questions when the sum is equal rather than greater or equal than, and it equals to a number other than 0, which is a simple stars and bars problem
A lot of mathematics is about recognising how to transform something you don't know how to solve into something you do know how to solve. If you know how to solve $$sum_i=1^4 x_i = k, x_i in -1,1$$ then can you substitute the solution into $$sum_k ge 0 sum_i=1^4 x_i = k, x_i in -1,1$$ to solve the original problem?
(Note: $k=0$ isn't really a special case after all).
answered Aug 20 at 10:02
Peter Taylor
7,82012239
answered Aug 20 at 10:02
Peter Taylor
7,82012239
answered Aug 20 at 10:02
Peter Taylor
7,82012239
answered Aug 20 at 10:02
Peter Taylor
7,82012239
7,82012239
add a comment |Â
add a comment |Â
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In the second inequality, $k_iin0,1$.
– Alexander Burstein
Aug 19 at 7:25