How can I calculate how many times a given subset of (finite) X will appear in the set of subsets of X of a...
$begingroup$
Let $X$ be a finite set. Suppose $R$ is some proper subset of X. Let $S$ be the set of subsets of $X$ of a given size; for example, $S$ could be the set of three-membered subsets of $X$. Is there any way to calculate how many members of $S$ have $R$ as a subset?
[edited this question due to some unclarities in the formulation]
combinatorics elementary-set-theory binomial-theorem
asked Jan 7 at 16:15
burt_gellorbsenburt_gellorbsen
61
$endgroup$
add a comment |
$begingroup$
Let $X$ be a finite set. Suppose $R$ is some proper subset of X. Let $S$ be the set of subsets of $X$ of a given size; for example, $S$ could be the set of three-membered subsets of $X$. Is there any way to calculate how many members of $S$ have $R$ as a subset?
[edited this question due to some unclarities in the formulation]
combinatorics elementary-set-theory binomial-theorem
asked Jan 7 at 16:15
burt_gellorbsenburt_gellorbsen
61
$endgroup$
1
$begingroup$
R will either appear or not appear in S depending on its size. Perhaps reclarify your question?
$endgroup$
– T. Fo
Jan 7 at 16:19
$begingroup$
Thanks for your quick response. I was assuming that when $|R| > |s|$ (where $s$ is a given member of $S$) then the number of times $R$ appears in $S$ would be zero. But I may be overlooking something.
$endgroup$
– burt_gellorbsen
Jan 7 at 16:21
$begingroup$
ya thats right. say S is the set of subsets of X of size n. then if |R| = n, R will be in S. otherwise it will not.
$endgroup$
– T. Fo
Jan 7 at 16:23
add a comment |
$begingroup$
Let $X$ be a finite set. Suppose $R$ is some proper subset of X. Let $S$ be the set of subsets of $X$ of a given size; for example, $S$ could be the set of three-membered subsets of $X$. Is there any way to calculate how many members of $S$ have $R$ as a subset?
[edited this question due to some unclarities in the formulation]
combinatorics elementary-set-theory binomial-theorem
asked Jan 7 at 16:15
burt_gellorbsenburt_gellorbsen
61
$endgroup$
Let $X$ be a finite set. Suppose $R$ is some proper subset of X. Let $S$ be the set of subsets of $X$ of a given size; for example, $S$ could be the set of three-membered subsets of $X$. Is there any way to calculate how many members of $S$ have $R$ as a subset?
[edited this question due to some unclarities in the formulation]
combinatorics elementary-set-theory binomial-theorem
combinatorics elementary-set-theory binomial-theorem
asked Jan 7 at 16:15
burt_gellorbsenburt_gellorbsen
61
asked Jan 7 at 16:15
burt_gellorbsenburt_gellorbsen
61
edited Jan 7 at 16:44
burt_gellorbsen
asked Jan 7 at 16:15
burt_gellorbsenburt_gellorbsen
61
asked Jan 7 at 16:15
burt_gellorbsenburt_gellorbsen
61
asked Jan 7 at 16:15
burt_gellorbsenburt_gellorbsen
61
61
1
$begingroup$
R will either appear or not appear in S depending on its size. Perhaps reclarify your question?
$endgroup$
– T. Fo
Jan 7 at 16:19
$begingroup$
Thanks for your quick response. I was assuming that when $|R| > |s|$ (where $s$ is a given member of $S$) then the number of times $R$ appears in $S$ would be zero. But I may be overlooking something.
$endgroup$
– burt_gellorbsen
Jan 7 at 16:21
$begingroup$
ya thats right. say S is the set of subsets of X of size n. then if |R| = n, R will be in S. otherwise it will not.
$endgroup$
– T. Fo
Jan 7 at 16:23
add a comment |
1
$begingroup$
R will either appear or not appear in S depending on its size. Perhaps reclarify your question?
$endgroup$
– T. Fo
Jan 7 at 16:19
$begingroup$
Thanks for your quick response. I was assuming that when $|R| > |s|$ (where $s$ is a given member of $S$) then the number of times $R$ appears in $S$ would be zero. But I may be overlooking something.
$endgroup$
– burt_gellorbsen
Jan 7 at 16:21
$begingroup$
ya thats right. say S is the set of subsets of X of size n. then if |R| = n, R will be in S. otherwise it will not.
$endgroup$
– T. Fo
Jan 7 at 16:23
1
1
$begingroup$
R will either appear or not appear in S depending on its size. Perhaps reclarify your question?
$endgroup$
– T. Fo
Jan 7 at 16:19
$begingroup$
R will either appear or not appear in S depending on its size. Perhaps reclarify your question?
$endgroup$
– T. Fo
Jan 7 at 16:19
$begingroup$
Thanks for your quick response. I was assuming that when $|R| > |s|$ (where $s$ is a given member of $S$) then the number of times $R$ appears in $S$ would be zero. But I may be overlooking something.
$endgroup$
– burt_gellorbsen
Jan 7 at 16:21
$begingroup$
Thanks for your quick response. I was assuming that when $|R| > |s|$ (where $s$ is a given member of $S$) then the number of times $R$ appears in $S$ would be zero. But I may be overlooking something.
$endgroup$
– burt_gellorbsen
Jan 7 at 16:21
$begingroup$
ya thats right. say S is the set of subsets of X of size n. then if |R| = n, R will be in S. otherwise it will not.
$endgroup$
– T. Fo
Jan 7 at 16:23
$begingroup$
ya thats right. say S is the set of subsets of X of size n. then if |R| = n, R will be in S. otherwise it will not.
$endgroup$
– T. Fo
Jan 7 at 16:23
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You rightly mentioned that if $|R|>|s|,sin S$, then the answer is $0$. On the other hand, if $|s|ge|R|$, you can count the number of subsets of $X$ of size $|s|$ with $R$ as a subset. For that, note that $|R|$ elements are already fixed; they are the elements of $R$. You can select the remaining $|s|-|R|$ elements from the remaining $|X|-|R|$ elements of $X$ in $displaystylebinom{|X|-|R|}{|s|-|R|}$ ways, which is the answer.
answered Jan 7 at 16:52
Shubham JohriShubham Johri
4,885717
$endgroup$
$begingroup$
This is great, thank you!
$endgroup$
– burt_gellorbsen
Jan 7 at 16:54
add a comment |
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1 Answer
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$begingroup$
You rightly mentioned that if $|R|>|s|,sin S$, then the answer is $0$. On the other hand, if $|s|ge|R|$, you can count the number of subsets of $X$ of size $|s|$ with $R$ as a subset. For that, note that $|R|$ elements are already fixed; they are the elements of $R$. You can select the remaining $|s|-|R|$ elements from the remaining $|X|-|R|$ elements of $X$ in $displaystylebinom{|X|-|R|}{|s|-|R|}$ ways, which is the answer.
answered Jan 7 at 16:52
Shubham JohriShubham Johri
4,885717
$endgroup$
$begingroup$
This is great, thank you!
$endgroup$
– burt_gellorbsen
Jan 7 at 16:54
add a comment |
$begingroup$
You rightly mentioned that if $|R|>|s|,sin S$, then the answer is $0$. On the other hand, if $|s|ge|R|$, you can count the number of subsets of $X$ of size $|s|$ with $R$ as a subset. For that, note that $|R|$ elements are already fixed; they are the elements of $R$. You can select the remaining $|s|-|R|$ elements from the remaining $|X|-|R|$ elements of $X$ in $displaystylebinom{|X|-|R|}{|s|-|R|}$ ways, which is the answer.
answered Jan 7 at 16:52
Shubham JohriShubham Johri
4,885717
$endgroup$
$begingroup$
This is great, thank you!
$endgroup$
– burt_gellorbsen
Jan 7 at 16:54
add a comment |
$begingroup$
You rightly mentioned that if $|R|>|s|,sin S$, then the answer is $0$. On the other hand, if $|s|ge|R|$, you can count the number of subsets of $X$ of size $|s|$ with $R$ as a subset. For that, note that $|R|$ elements are already fixed; they are the elements of $R$. You can select the remaining $|s|-|R|$ elements from the remaining $|X|-|R|$ elements of $X$ in $displaystylebinom{|X|-|R|}{|s|-|R|}$ ways, which is the answer.
answered Jan 7 at 16:52
Shubham JohriShubham Johri
4,885717
$endgroup$
You rightly mentioned that if $|R|>|s|,sin S$, then the answer is $0$. On the other hand, if $|s|ge|R|$, you can count the number of subsets of $X$ of size $|s|$ with $R$ as a subset. For that, note that $|R|$ elements are already fixed; they are the elements of $R$. You can select the remaining $|s|-|R|$ elements from the remaining $|X|-|R|$ elements of $X$ in $displaystylebinom{|X|-|R|}{|s|-|R|}$ ways, which is the answer.
answered Jan 7 at 16:52
Shubham JohriShubham Johri
4,885717
answered Jan 7 at 16:52
Shubham JohriShubham Johri
4,885717
answered Jan 7 at 16:52
Shubham JohriShubham Johri
4,885717
answered Jan 7 at 16:52
Shubham JohriShubham Johri
4,885717
4,885717
$begingroup$
This is great, thank you!
$endgroup$
– burt_gellorbsen
Jan 7 at 16:54
add a comment |
$begingroup$
This is great, thank you!
$endgroup$
– burt_gellorbsen
Jan 7 at 16:54
$begingroup$
This is great, thank you!
$endgroup$
– burt_gellorbsen
Jan 7 at 16:54
$begingroup$
This is great, thank you!
$endgroup$
– burt_gellorbsen
Jan 7 at 16:54
add a comment |
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1
$begingroup$
R will either appear or not appear in S depending on its size. Perhaps reclarify your question?
$endgroup$
– T. Fo
Jan 7 at 16:19
$begingroup$
Thanks for your quick response. I was assuming that when $|R| > |s|$ (where $s$ is a given member of $S$) then the number of times $R$ appears in $S$ would be zero. But I may be overlooking something.
$endgroup$
– burt_gellorbsen
Jan 7 at 16:21
$begingroup$
ya thats right. say S is the set of subsets of X of size n. then if |R| = n, R will be in S. otherwise it will not.
$endgroup$
– T. Fo
Jan 7 at 16:23