Mixing
Calculating number of times a given element appears in powerset subsets of a particular size.
$begingroup$
Let $X$ be a finite set, and $P(X)$ the power set of $X$. I understand from here how to calculate the number of subsets of a particular size in $P(X)$. For a given member of $X$ (call it $n$) how can I calculate the number of times $n$ appears in a set of subsets of a given size?
For example, suppose $X = {1, 2, 3}$ and $n = 2$. Then the subsets of $P(X)$ with two members are ${1, 2}, {1, 3}$ and ${2, 3}$. $n$ is in two thirds of these. But is there a more general way for determining the number of times $n$ appears in a subset of $P(X)$ containing only sets of identical size?
combinatorics elementary-set-theory binomial-theorem
asked Jan 7 at 14:52
burt_gellorbsenburt_gellorbsen
61
$endgroup$
add a comment |
$begingroup$
Let $X$ be a finite set, and $P(X)$ the power set of $X$. I understand from here how to calculate the number of subsets of a particular size in $P(X)$. For a given member of $X$ (call it $n$) how can I calculate the number of times $n$ appears in a set of subsets of a given size?
For example, suppose $X = {1, 2, 3}$ and $n = 2$. Then the subsets of $P(X)$ with two members are ${1, 2}, {1, 3}$ and ${2, 3}$. $n$ is in two thirds of these. But is there a more general way for determining the number of times $n$ appears in a subset of $P(X)$ containing only sets of identical size?
combinatorics elementary-set-theory binomial-theorem
asked Jan 7 at 14:52
burt_gellorbsenburt_gellorbsen
61
$endgroup$
add a comment |
$begingroup$
Let $X$ be a finite set, and $P(X)$ the power set of $X$. I understand from here how to calculate the number of subsets of a particular size in $P(X)$. For a given member of $X$ (call it $n$) how can I calculate the number of times $n$ appears in a set of subsets of a given size?
For example, suppose $X = {1, 2, 3}$ and $n = 2$. Then the subsets of $P(X)$ with two members are ${1, 2}, {1, 3}$ and ${2, 3}$. $n$ is in two thirds of these. But is there a more general way for determining the number of times $n$ appears in a subset of $P(X)$ containing only sets of identical size?
combinatorics elementary-set-theory binomial-theorem
asked Jan 7 at 14:52
burt_gellorbsenburt_gellorbsen
61
$endgroup$
Let $X$ be a finite set, and $P(X)$ the power set of $X$. I understand from here how to calculate the number of subsets of a particular size in $P(X)$. For a given member of $X$ (call it $n$) how can I calculate the number of times $n$ appears in a set of subsets of a given size?
For example, suppose $X = {1, 2, 3}$ and $n = 2$. Then the subsets of $P(X)$ with two members are ${1, 2}, {1, 3}$ and ${2, 3}$. $n$ is in two thirds of these. But is there a more general way for determining the number of times $n$ appears in a subset of $P(X)$ containing only sets of identical size?
combinatorics elementary-set-theory binomial-theorem
combinatorics elementary-set-theory binomial-theorem
asked Jan 7 at 14:52
burt_gellorbsenburt_gellorbsen
61
asked Jan 7 at 14:52
burt_gellorbsenburt_gellorbsen
61
asked Jan 7 at 14:52
burt_gellorbsenburt_gellorbsen
61
asked Jan 7 at 14:52
burt_gellorbsenburt_gellorbsen
61
asked Jan 7 at 14:52
burt_gellorbsenburt_gellorbsen
61
61
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Suppose $X$ has $N$ elements. The probability that a particular element (you call it $n$) is in a randomly chosen
subset of size $k$ is just $k/N$.
So if there are $M$ subsets of size $k$ the number of them that contain $n$ is $(k/N)M$.
Since you say you know how to calculate $M$ you're done.
answered Jan 7 at 15:08
Ethan BolkerEthan Bolker
42.4k549112
$endgroup$
$begingroup$
Thanks so much, really appreciate this. I've just realised that I slightly misstated my question, which was in fact: If $n subseteq X$, how many members of the set of k-sized subsets (e.g. the set ${k_1, k_2 ...}$ are such that $n subseteq$ those members? You seem like a much more experienced stack user than me -- should I edit my existing question or am I supposed to add a new one fresh?
$endgroup$
– burt_gellorbsen
Jan 7 at 15:15
$begingroup$
@burt_gellorbsen Normally I'd suggest accepting an answer to this one and asking a new correct one. In this case you might be able to work out the probability in the first paragraph of my answer for your modified question, then continue as in the rest of the answer.
$endgroup$
– Ethan Bolker
Jan 7 at 15:22
add a comment |
$begingroup$
HINT- Let the size of the desired subsets be $k$ and the total size of the set $N$. So you want the number of $k$-subsets of the set which have a particular element $n$. Now let's assume element $n$ is present in a subset. That leaves $k-1$ elements inside the subset that have to be filled from the remaining $N-1$. The number of ways to select $k-1$ elements from a group of $N-1$ is ....
EDIT- Since the OP insisted, I'm giving out the answer
$$binom{N-1}{k-1}$$
answered Jan 7 at 15:05
Sauhard SharmaSauhard Sharma
953318
$endgroup$
$begingroup$
I really appreciate your response. I'm afraid you will have to spell out the answer for me though --- sorry!
$endgroup$
– burt_gellorbsen
Jan 7 at 15:06
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Suppose $X$ has $N$ elements. The probability that a particular element (you call it $n$) is in a randomly chosen
subset of size $k$ is just $k/N$.
So if there are $M$ subsets of size $k$ the number of them that contain $n$ is $(k/N)M$.
Since you say you know how to calculate $M$ you're done.
answered Jan 7 at 15:08
Ethan BolkerEthan Bolker
42.4k549112
$endgroup$
$begingroup$
Thanks so much, really appreciate this. I've just realised that I slightly misstated my question, which was in fact: If $n subseteq X$, how many members of the set of k-sized subsets (e.g. the set ${k_1, k_2 ...}$ are such that $n subseteq$ those members? You seem like a much more experienced stack user than me -- should I edit my existing question or am I supposed to add a new one fresh?
$endgroup$
– burt_gellorbsen
Jan 7 at 15:15
$begingroup$
@burt_gellorbsen Normally I'd suggest accepting an answer to this one and asking a new correct one. In this case you might be able to work out the probability in the first paragraph of my answer for your modified question, then continue as in the rest of the answer.
$endgroup$
– Ethan Bolker
Jan 7 at 15:22
add a comment |
$begingroup$
Suppose $X$ has $N$ elements. The probability that a particular element (you call it $n$) is in a randomly chosen
subset of size $k$ is just $k/N$.
So if there are $M$ subsets of size $k$ the number of them that contain $n$ is $(k/N)M$.
Since you say you know how to calculate $M$ you're done.
answered Jan 7 at 15:08
Ethan BolkerEthan Bolker
42.4k549112
$endgroup$
$begingroup$
Thanks so much, really appreciate this. I've just realised that I slightly misstated my question, which was in fact: If $n subseteq X$, how many members of the set of k-sized subsets (e.g. the set ${k_1, k_2 ...}$ are such that $n subseteq$ those members? You seem like a much more experienced stack user than me -- should I edit my existing question or am I supposed to add a new one fresh?
$endgroup$
– burt_gellorbsen
Jan 7 at 15:15
$begingroup$
@burt_gellorbsen Normally I'd suggest accepting an answer to this one and asking a new correct one. In this case you might be able to work out the probability in the first paragraph of my answer for your modified question, then continue as in the rest of the answer.
$endgroup$
– Ethan Bolker
Jan 7 at 15:22
add a comment |
$begingroup$
Suppose $X$ has $N$ elements. The probability that a particular element (you call it $n$) is in a randomly chosen
subset of size $k$ is just $k/N$.
So if there are $M$ subsets of size $k$ the number of them that contain $n$ is $(k/N)M$.
Since you say you know how to calculate $M$ you're done.
answered Jan 7 at 15:08
Ethan BolkerEthan Bolker
42.4k549112
$endgroup$
Suppose $X$ has $N$ elements. The probability that a particular element (you call it $n$) is in a randomly chosen
subset of size $k$ is just $k/N$.
So if there are $M$ subsets of size $k$ the number of them that contain $n$ is $(k/N)M$.
Since you say you know how to calculate $M$ you're done.
answered Jan 7 at 15:08
Ethan BolkerEthan Bolker
42.4k549112
answered Jan 7 at 15:08
Ethan BolkerEthan Bolker
42.4k549112
answered Jan 7 at 15:08
Ethan BolkerEthan Bolker
42.4k549112
answered Jan 7 at 15:08
Ethan BolkerEthan Bolker
42.4k549112
42.4k549112
$begingroup$
Thanks so much, really appreciate this. I've just realised that I slightly misstated my question, which was in fact: If $n subseteq X$, how many members of the set of k-sized subsets (e.g. the set ${k_1, k_2 ...}$ are such that $n subseteq$ those members? You seem like a much more experienced stack user than me -- should I edit my existing question or am I supposed to add a new one fresh?
$endgroup$
– burt_gellorbsen
Jan 7 at 15:15
$begingroup$
@burt_gellorbsen Normally I'd suggest accepting an answer to this one and asking a new correct one. In this case you might be able to work out the probability in the first paragraph of my answer for your modified question, then continue as in the rest of the answer.
$endgroup$
– Ethan Bolker
Jan 7 at 15:22
add a comment |
$begingroup$
Thanks so much, really appreciate this. I've just realised that I slightly misstated my question, which was in fact: If $n subseteq X$, how many members of the set of k-sized subsets (e.g. the set ${k_1, k_2 ...}$ are such that $n subseteq$ those members? You seem like a much more experienced stack user than me -- should I edit my existing question or am I supposed to add a new one fresh?
$endgroup$
– burt_gellorbsen
Jan 7 at 15:15
$begingroup$
@burt_gellorbsen Normally I'd suggest accepting an answer to this one and asking a new correct one. In this case you might be able to work out the probability in the first paragraph of my answer for your modified question, then continue as in the rest of the answer.
$endgroup$
– Ethan Bolker
Jan 7 at 15:22
$begingroup$
Thanks so much, really appreciate this. I've just realised that I slightly misstated my question, which was in fact: If $n subseteq X$, how many members of the set of k-sized subsets (e.g. the set ${k_1, k_2 ...}$ are such that $n subseteq$ those members? You seem like a much more experienced stack user than me -- should I edit my existing question or am I supposed to add a new one fresh?
$endgroup$
– burt_gellorbsen
Jan 7 at 15:15
$begingroup$
Thanks so much, really appreciate this. I've just realised that I slightly misstated my question, which was in fact: If $n subseteq X$, how many members of the set of k-sized subsets (e.g. the set ${k_1, k_2 ...}$ are such that $n subseteq$ those members? You seem like a much more experienced stack user than me -- should I edit my existing question or am I supposed to add a new one fresh?
$endgroup$
– burt_gellorbsen
Jan 7 at 15:15
$begingroup$
@burt_gellorbsen Normally I'd suggest accepting an answer to this one and asking a new correct one. In this case you might be able to work out the probability in the first paragraph of my answer for your modified question, then continue as in the rest of the answer.
$endgroup$
– Ethan Bolker
Jan 7 at 15:22
$begingroup$
@burt_gellorbsen Normally I'd suggest accepting an answer to this one and asking a new correct one. In this case you might be able to work out the probability in the first paragraph of my answer for your modified question, then continue as in the rest of the answer.
$endgroup$
– Ethan Bolker
Jan 7 at 15:22
add a comment |
$begingroup$
HINT- Let the size of the desired subsets be $k$ and the total size of the set $N$. So you want the number of $k$-subsets of the set which have a particular element $n$. Now let's assume element $n$ is present in a subset. That leaves $k-1$ elements inside the subset that have to be filled from the remaining $N-1$. The number of ways to select $k-1$ elements from a group of $N-1$ is ....
EDIT- Since the OP insisted, I'm giving out the answer
$$binom{N-1}{k-1}$$
answered Jan 7 at 15:05
Sauhard SharmaSauhard Sharma
953318
$endgroup$
$begingroup$
I really appreciate your response. I'm afraid you will have to spell out the answer for me though --- sorry!
$endgroup$
– burt_gellorbsen
Jan 7 at 15:06
add a comment |
$begingroup$
HINT- Let the size of the desired subsets be $k$ and the total size of the set $N$. So you want the number of $k$-subsets of the set which have a particular element $n$. Now let's assume element $n$ is present in a subset. That leaves $k-1$ elements inside the subset that have to be filled from the remaining $N-1$. The number of ways to select $k-1$ elements from a group of $N-1$ is ....
EDIT- Since the OP insisted, I'm giving out the answer
$$binom{N-1}{k-1}$$
answered Jan 7 at 15:05
Sauhard SharmaSauhard Sharma
953318
$endgroup$
$begingroup$
I really appreciate your response. I'm afraid you will have to spell out the answer for me though --- sorry!
$endgroup$
– burt_gellorbsen
Jan 7 at 15:06
add a comment |
$begingroup$
HINT- Let the size of the desired subsets be $k$ and the total size of the set $N$. So you want the number of $k$-subsets of the set which have a particular element $n$. Now let's assume element $n$ is present in a subset. That leaves $k-1$ elements inside the subset that have to be filled from the remaining $N-1$. The number of ways to select $k-1$ elements from a group of $N-1$ is ....
EDIT- Since the OP insisted, I'm giving out the answer
$$binom{N-1}{k-1}$$
answered Jan 7 at 15:05
Sauhard SharmaSauhard Sharma
953318
$endgroup$
HINT- Let the size of the desired subsets be $k$ and the total size of the set $N$. So you want the number of $k$-subsets of the set which have a particular element $n$. Now let's assume element $n$ is present in a subset. That leaves $k-1$ elements inside the subset that have to be filled from the remaining $N-1$. The number of ways to select $k-1$ elements from a group of $N-1$ is ....
EDIT- Since the OP insisted, I'm giving out the answer
$$binom{N-1}{k-1}$$
answered Jan 7 at 15:05
Sauhard SharmaSauhard Sharma
953318
edited Jan 7 at 15:09
answered Jan 7 at 15:05
Sauhard SharmaSauhard Sharma
953318
answered Jan 7 at 15:05
Sauhard SharmaSauhard Sharma
953318
answered Jan 7 at 15:05
Sauhard SharmaSauhard Sharma
953318
953318
$begingroup$
I really appreciate your response. I'm afraid you will have to spell out the answer for me though --- sorry!
$endgroup$
– burt_gellorbsen
Jan 7 at 15:06
add a comment |
$begingroup$
I really appreciate your response. I'm afraid you will have to spell out the answer for me though --- sorry!
$endgroup$
– burt_gellorbsen
Jan 7 at 15:06
$begingroup$
I really appreciate your response. I'm afraid you will have to spell out the answer for me though --- sorry!
$endgroup$
– burt_gellorbsen
Jan 7 at 15:06
$begingroup$
I really appreciate your response. I'm afraid you will have to spell out the answer for me though --- sorry!
$endgroup$
– burt_gellorbsen
Jan 7 at 15:06
add a comment |
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