How can I calculate how many times a given subset of (finite) X will appear in the set of subsets of X of a...












0












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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







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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
















0












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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














0












0








0





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 7 at 16:44







burt_gellorbsen

















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














  • 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










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.






share|cite|improve this answer









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  • $begingroup$
    This is great, thank you!
    $endgroup$
    – burt_gellorbsen
    Jan 7 at 16:54











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1 Answer
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1 Answer
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active

oldest

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active

oldest

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active

oldest

votes









1












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This is great, thank you!
    $endgroup$
    – burt_gellorbsen
    Jan 7 at 16:54
















1












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This is great, thank you!
    $endgroup$
    – burt_gellorbsen
    Jan 7 at 16:54














1












1








1





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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


















  • $begingroup$
    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


















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