In how many different ways can I arrange $n$ girls and $m$ boys with no at least $k$ boys or $k$ girls sit...
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For example, we have $n = 5$ boys and $m = 6$ girls, we don't want $k le min(n, m)$ | $k = 3$ boys or girls to sit next to each other.
A valid arrangement: $BBGGBGGBGG$
An invalid arrangement: $BBBGGBGGBGG$ - There are $3$ boys sit next to each other.
And we ignore the permutation between any group of boys and girls so $BBGGBGGBGG$ is considered only $1$ way
combinatorics permutations
asked Nov 18 at 9:53
Nhân Nguyễn
113
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show 2 more comments
up vote
2
down vote
favorite
For example, we have $n = 5$ boys and $m = 6$ girls, we don't want $k le min(n, m)$ | $k = 3$ boys or girls to sit next to each other.
A valid arrangement: $BBGGBGGBGG$
An invalid arrangement: $BBBGGBGGBGG$ - There are $3$ boys sit next to each other.
And we ignore the permutation between any group of boys and girls so $BBGGBGGBGG$ is considered only $1$ way
combinatorics permutations
asked Nov 18 at 9:53
Nhân Nguyễn
113
elcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:58
Note that sometimes it is impossible, for example if you have 10 boys and 2 girls and k=3, because in any arrangement you use there will be 3 or more boys sitting next to each other. I think for the general case you must have $k>frac{|m-n|}{2}-2$ But I might be wrong in this one
– Fareed AF
Nov 18 at 10:20
I'm sorry, It should be like this $k le min(n, m)$.
– Nhân Nguyễn
Nov 18 at 10:22
@FareedAF If that scenario happens, the answer can only be 0 because there are no way we can arrange, right?
– Nhân Nguyễn
Nov 18 at 10:32
Yes thats what I meant by impossible, 0 ways
– Fareed AF
Nov 18 at 10:33
|
show 2 more comments
up vote
2
down vote
favorite
up vote
2
down vote
favorite
For example, we have $n = 5$ boys and $m = 6$ girls, we don't want $k le min(n, m)$ | $k = 3$ boys or girls to sit next to each other.
A valid arrangement: $BBGGBGGBGG$
An invalid arrangement: $BBBGGBGGBGG$ - There are $3$ boys sit next to each other.
And we ignore the permutation between any group of boys and girls so $BBGGBGGBGG$ is considered only $1$ way
combinatorics permutations
asked Nov 18 at 9:53
Nhân Nguyễn
113
For example, we have $n = 5$ boys and $m = 6$ girls, we don't want $k le min(n, m)$ | $k = 3$ boys or girls to sit next to each other.
A valid arrangement: $BBGGBGGBGG$
An invalid arrangement: $BBBGGBGGBGG$ - There are $3$ boys sit next to each other.
And we ignore the permutation between any group of boys and girls so $BBGGBGGBGG$ is considered only $1$ way
combinatorics permutations
combinatorics permutations
asked Nov 18 at 9:53
Nhân Nguyễn
113
asked Nov 18 at 9:53
Nhân Nguyễn
113
edited Nov 18 at 10:24
asked Nov 18 at 9:53
Nhân Nguyễn
113
asked Nov 18 at 9:53
Nhân Nguyễn
113
asked Nov 18 at 9:53
Nhân Nguyễn
113
113
elcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:58
Note that sometimes it is impossible, for example if you have 10 boys and 2 girls and k=3, because in any arrangement you use there will be 3 or more boys sitting next to each other. I think for the general case you must have $k>frac{|m-n|}{2}-2$ But I might be wrong in this one
– Fareed AF
Nov 18 at 10:20
I'm sorry, It should be like this $k le min(n, m)$.
– Nhân Nguyễn
Nov 18 at 10:22
@FareedAF If that scenario happens, the answer can only be 0 because there are no way we can arrange, right?
– Nhân Nguyễn
Nov 18 at 10:32
Yes thats what I meant by impossible, 0 ways
– Fareed AF
Nov 18 at 10:33
|
show 2 more comments
elcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:58
Note that sometimes it is impossible, for example if you have 10 boys and 2 girls and k=3, because in any arrangement you use there will be 3 or more boys sitting next to each other. I think for the general case you must have $k>frac{|m-n|}{2}-2$ But I might be wrong in this one
– Fareed AF
Nov 18 at 10:20
I'm sorry, It should be like this $k le min(n, m)$.
– Nhân Nguyễn
Nov 18 at 10:22
@FareedAF If that scenario happens, the answer can only be 0 because there are no way we can arrange, right?
– Nhân Nguyễn
Nov 18 at 10:32
Yes thats what I meant by impossible, 0 ways
– Fareed AF
Nov 18 at 10:33
elcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:58
elcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:58
Note that sometimes it is impossible, for example if you have 10 boys and 2 girls and k=3, because in any arrangement you use there will be 3 or more boys sitting next to each other. I think for the general case you must have $k>frac{|m-n|}{2}-2$ But I might be wrong in this one
– Fareed AF
Nov 18 at 10:20
Note that sometimes it is impossible, for example if you have 10 boys and 2 girls and k=3, because in any arrangement you use there will be 3 or more boys sitting next to each other. I think for the general case you must have $k>frac{|m-n|}{2}-2$ But I might be wrong in this one
– Fareed AF
Nov 18 at 10:20
I'm sorry, It should be like this $k le min(n, m)$.
– Nhân Nguyễn
Nov 18 at 10:22
I'm sorry, It should be like this $k le min(n, m)$.
– Nhân Nguyễn
Nov 18 at 10:22
@FareedAF If that scenario happens, the answer can only be 0 because there are no way we can arrange, right?
– Nhân Nguyễn
Nov 18 at 10:32
@FareedAF If that scenario happens, the answer can only be 0 because there are no way we can arrange, right?
– Nhân Nguyễn
Nov 18 at 10:32
Yes thats what I meant by impossible, 0 ways
– Fareed AF
Nov 18 at 10:33
Yes thats what I meant by impossible, 0 ways
– Fareed AF
Nov 18 at 10:33
|
show 2 more comments
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elcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:58
Note that sometimes it is impossible, for example if you have 10 boys and 2 girls and k=3, because in any arrangement you use there will be 3 or more boys sitting next to each other. I think for the general case you must have $k>frac{|m-n|}{2}-2$ But I might be wrong in this one
– Fareed AF
Nov 18 at 10:20
I'm sorry, It should be like this $k le min(n, m)$.
– Nhân Nguyễn
Nov 18 at 10:22
@FareedAF If that scenario happens, the answer can only be 0 because there are no way we can arrange, right?
– Nhân Nguyễn
Nov 18 at 10:32
Yes thats what I meant by impossible, 0 ways
– Fareed AF
Nov 18 at 10:33