In how many ways can the entries of a $3 times 3$ table be filled in with $0$s and $1$s so that the sum of...
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In how many ways can the entries of a $3 times 3$ table be filled in with the numbers zero and one so that the sum of numbers in at least one of the rows is equal to zero?
combinatorics
edited Dec 20 '18 at 0:57
N. F. Taussig
44k93356
asked Dec 20 '18 at 0:17
rasool sadatrasool sadat
113
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closed as unclear what you're asking by amWhy, Shailesh, Karn Watcharasupat, KReiser, Carl Schildkraut Dec 20 '18 at 6:06
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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$begingroup$
In how many ways can the entries of a $3 times 3$ table be filled in with the numbers zero and one so that the sum of numbers in at least one of the rows is equal to zero?
combinatorics
edited Dec 20 '18 at 0:57
N. F. Taussig
44k93356
asked Dec 20 '18 at 0:17
rasool sadatrasool sadat
113
$endgroup$
closed as unclear what you're asking by amWhy, Shailesh, Karn Watcharasupat, KReiser, Carl Schildkraut Dec 20 '18 at 6:06
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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Yes: you can put zero in all of them, for instance.
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– Saucy O'Path
Dec 20 '18 at 0:30
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"so that the sum of numbers" so that the sum of numbers does what? If you fill them all with $1$ you can avoid this. If you fill the first row with $0$ you can do this. What actually is the question?
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– fleablood
Dec 20 '18 at 0:39
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For the sum of a row to add up to zero all the houses must be $0$. I dont see any reason why filling a row with all $0$s would be at all difficult. Just let the first row be $0$ and the rest can be any combination you want.
$endgroup$
– fleablood
Dec 20 '18 at 0:41
add a comment |
$begingroup$
In how many ways can the entries of a $3 times 3$ table be filled in with the numbers zero and one so that the sum of numbers in at least one of the rows is equal to zero?
combinatorics
edited Dec 20 '18 at 0:57
N. F. Taussig
44k93356
asked Dec 20 '18 at 0:17
rasool sadatrasool sadat
113
$endgroup$
In how many ways can the entries of a $3 times 3$ table be filled in with the numbers zero and one so that the sum of numbers in at least one of the rows is equal to zero?
combinatorics
combinatorics
edited Dec 20 '18 at 0:57
N. F. Taussig
44k93356
asked Dec 20 '18 at 0:17
rasool sadatrasool sadat
113
edited Dec 20 '18 at 0:57
N. F. Taussig
44k93356
asked Dec 20 '18 at 0:17
rasool sadatrasool sadat
113
edited Dec 20 '18 at 0:57
N. F. Taussig
44k93356
edited Dec 20 '18 at 0:57
N. F. Taussig
44k93356
edited Dec 20 '18 at 0:57
N. F. Taussig
44k93356
44k93356
asked Dec 20 '18 at 0:17
rasool sadatrasool sadat
113
asked Dec 20 '18 at 0:17
rasool sadatrasool sadat
113
asked Dec 20 '18 at 0:17
rasool sadatrasool sadat
113
113
closed as unclear what you're asking by amWhy, Shailesh, Karn Watcharasupat, KReiser, Carl Schildkraut Dec 20 '18 at 6:06
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by amWhy, Shailesh, Karn Watcharasupat, KReiser, Carl Schildkraut Dec 20 '18 at 6:06
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Yes: you can put zero in all of them, for instance.
$endgroup$
– Saucy O'Path
Dec 20 '18 at 0:30
$begingroup$
"so that the sum of numbers" so that the sum of numbers does what? If you fill them all with $1$ you can avoid this. If you fill the first row with $0$ you can do this. What actually is the question?
$endgroup$
– fleablood
Dec 20 '18 at 0:39
$begingroup$
For the sum of a row to add up to zero all the houses must be $0$. I dont see any reason why filling a row with all $0$s would be at all difficult. Just let the first row be $0$ and the rest can be any combination you want.
$endgroup$
– fleablood
Dec 20 '18 at 0:41
add a comment |
$begingroup$
Yes: you can put zero in all of them, for instance.
$endgroup$
– Saucy O'Path
Dec 20 '18 at 0:30
$begingroup$
"so that the sum of numbers" so that the sum of numbers does what? If you fill them all with $1$ you can avoid this. If you fill the first row with $0$ you can do this. What actually is the question?
$endgroup$
– fleablood
Dec 20 '18 at 0:39
$begingroup$
For the sum of a row to add up to zero all the houses must be $0$. I dont see any reason why filling a row with all $0$s would be at all difficult. Just let the first row be $0$ and the rest can be any combination you want.
$endgroup$
– fleablood
Dec 20 '18 at 0:41
$begingroup$
Yes: you can put zero in all of them, for instance.
$endgroup$
– Saucy O'Path
Dec 20 '18 at 0:30
$begingroup$
Yes: you can put zero in all of them, for instance.
$endgroup$
– Saucy O'Path
Dec 20 '18 at 0:30
$begingroup$
"so that the sum of numbers" so that the sum of numbers does what? If you fill them all with $1$ you can avoid this. If you fill the first row with $0$ you can do this. What actually is the question?
$endgroup$
– fleablood
Dec 20 '18 at 0:39
$begingroup$
"so that the sum of numbers" so that the sum of numbers does what? If you fill them all with $1$ you can avoid this. If you fill the first row with $0$ you can do this. What actually is the question?
$endgroup$
– fleablood
Dec 20 '18 at 0:39
$begingroup$
For the sum of a row to add up to zero all the houses must be $0$. I dont see any reason why filling a row with all $0$s would be at all difficult. Just let the first row be $0$ and the rest can be any combination you want.
$endgroup$
– fleablood
Dec 20 '18 at 0:41
$begingroup$
For the sum of a row to add up to zero all the houses must be $0$. I dont see any reason why filling a row with all $0$s would be at all difficult. Just let the first row be $0$ and the rest can be any combination you want.
$endgroup$
– fleablood
Dec 20 '18 at 0:41
add a comment |
1 Answer
1
active
oldest
votes
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Hint: Use inclusion-exclusion over the sets $A_1,A_2,A_3$ where $A_i$ is the set "The ways to fill a $3times 3$ table with $0$'s and $1$'s such that the sum of the numbers in the $i$'th row is zero."
You are (presumably after correcting your grammar) trying to calculate $|A_1cup A_2cup A_3|$
Additional hint:
Inclusion-exclusion implies $|Acup Bcup C|=|A|+|B|+|C|-|Acap B|-|Acap C|-|Bcap C|+|Acap Bcap C|$
Additional hint:
If the sum of the first row is zero, then every entry in the first row is zero. Then in counting how many ways we can fill the rest of the table, we have $2$ options for the secondrow-firstcolumn entry, $2$ options for the secondrow-secondcolumn entry, and so on up until $2$ options for the thirdrow-thirdcolumn entry. Applying rule of product yields a count.
answered Dec 20 '18 at 0:34
JMoravitzJMoravitz
47k33886
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: Use inclusion-exclusion over the sets $A_1,A_2,A_3$ where $A_i$ is the set "The ways to fill a $3times 3$ table with $0$'s and $1$'s such that the sum of the numbers in the $i$'th row is zero."
You are (presumably after correcting your grammar) trying to calculate $|A_1cup A_2cup A_3|$
Additional hint:
Inclusion-exclusion implies $|Acup Bcup C|=|A|+|B|+|C|-|Acap B|-|Acap C|-|Bcap C|+|Acap Bcap C|$
Additional hint:
If the sum of the first row is zero, then every entry in the first row is zero. Then in counting how many ways we can fill the rest of the table, we have $2$ options for the secondrow-firstcolumn entry, $2$ options for the secondrow-secondcolumn entry, and so on up until $2$ options for the thirdrow-thirdcolumn entry. Applying rule of product yields a count.
answered Dec 20 '18 at 0:34
JMoravitzJMoravitz
47k33886
$endgroup$
add a comment |
$begingroup$
Hint: Use inclusion-exclusion over the sets $A_1,A_2,A_3$ where $A_i$ is the set "The ways to fill a $3times 3$ table with $0$'s and $1$'s such that the sum of the numbers in the $i$'th row is zero."
You are (presumably after correcting your grammar) trying to calculate $|A_1cup A_2cup A_3|$
Additional hint:
Inclusion-exclusion implies $|Acup Bcup C|=|A|+|B|+|C|-|Acap B|-|Acap C|-|Bcap C|+|Acap Bcap C|$
Additional hint:
If the sum of the first row is zero, then every entry in the first row is zero. Then in counting how many ways we can fill the rest of the table, we have $2$ options for the secondrow-firstcolumn entry, $2$ options for the secondrow-secondcolumn entry, and so on up until $2$ options for the thirdrow-thirdcolumn entry. Applying rule of product yields a count.
answered Dec 20 '18 at 0:34
JMoravitzJMoravitz
47k33886
$endgroup$
add a comment |
$begingroup$
Hint: Use inclusion-exclusion over the sets $A_1,A_2,A_3$ where $A_i$ is the set "The ways to fill a $3times 3$ table with $0$'s and $1$'s such that the sum of the numbers in the $i$'th row is zero."
You are (presumably after correcting your grammar) trying to calculate $|A_1cup A_2cup A_3|$
Additional hint:
Inclusion-exclusion implies $|Acup Bcup C|=|A|+|B|+|C|-|Acap B|-|Acap C|-|Bcap C|+|Acap Bcap C|$
Additional hint:
If the sum of the first row is zero, then every entry in the first row is zero. Then in counting how many ways we can fill the rest of the table, we have $2$ options for the secondrow-firstcolumn entry, $2$ options for the secondrow-secondcolumn entry, and so on up until $2$ options for the thirdrow-thirdcolumn entry. Applying rule of product yields a count.
answered Dec 20 '18 at 0:34
JMoravitzJMoravitz
47k33886
$endgroup$
Hint: Use inclusion-exclusion over the sets $A_1,A_2,A_3$ where $A_i$ is the set "The ways to fill a $3times 3$ table with $0$'s and $1$'s such that the sum of the numbers in the $i$'th row is zero."
You are (presumably after correcting your grammar) trying to calculate $|A_1cup A_2cup A_3|$
Additional hint:
Inclusion-exclusion implies $|Acup Bcup C|=|A|+|B|+|C|-|Acap B|-|Acap C|-|Bcap C|+|Acap Bcap C|$
Additional hint:
If the sum of the first row is zero, then every entry in the first row is zero. Then in counting how many ways we can fill the rest of the table, we have $2$ options for the secondrow-firstcolumn entry, $2$ options for the secondrow-secondcolumn entry, and so on up until $2$ options for the thirdrow-thirdcolumn entry. Applying rule of product yields a count.
answered Dec 20 '18 at 0:34
JMoravitzJMoravitz
47k33886
answered Dec 20 '18 at 0:34
JMoravitzJMoravitz
47k33886
answered Dec 20 '18 at 0:34
JMoravitzJMoravitz
47k33886
answered Dec 20 '18 at 0:34
JMoravitzJMoravitz
47k33886
47k33886
add a comment |
add a comment |
$begingroup$
Yes: you can put zero in all of them, for instance.
$endgroup$
– Saucy O'Path
Dec 20 '18 at 0:30
$begingroup$
"so that the sum of numbers" so that the sum of numbers does what? If you fill them all with $1$ you can avoid this. If you fill the first row with $0$ you can do this. What actually is the question?
$endgroup$
– fleablood
Dec 20 '18 at 0:39
$begingroup$
For the sum of a row to add up to zero all the houses must be $0$. I dont see any reason why filling a row with all $0$s would be at all difficult. Just let the first row be $0$ and the rest can be any combination you want.
$endgroup$
– fleablood
Dec 20 '18 at 0:41