How many subsets of ${1,2,3,ldots,100}$ contain all the even numbers? The 2019 Stack Overflow...
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How many subsets of ${1,2,3,ldots,100}$ contain all the even numbers?
The 2019 Stack Overflow Developer Survey Results Are InHow to count number of pairs of subsets $(A,B)$ of $ X={1,2,..,100}$ under the given constraint?How many selections will contain the element?How many 3-subsets of ${1,2,ldots,10}$ contain at least two consecutive integers?How many non empty subsets of {1, 2, …, n} satisfy that the sum of their elements is even?How many subsets of $S$ are there that contain $x$ but do not contain $y$?$4$-element subsets of the set ${1,2,3,ldots,10}$ that do not contain any pair of consecutive numbersA set contains ${1,2,3,4,5…n}$ where $n$ is a even number. how many subsets that contain only even numbers are there$?$Possible number of subsets that must intersect of ${1,2,3,…100}$How many subsets of set {1,2,…,10} contain at least one odd integer?How many subsets are there such that their pairwise intersection is empty?
$begingroup$
Problem. How many subsets of ${1,2,3,ldots,100}$ contain all the even numbers?
I am not sure what is meant by "contain all the even numbers". If we are talking about all the even numbers between $1$ and $100$ (including $100$), then there is only one subset but if we are talking about all subsets whose elements are only even numbers, then since there are $50$ even numbers between $1$ and $100$ (including $100$), then there are $2^{50}-1$ subsets of the original set whose elements are only even numbers and the $-1$ is just for subtracting the empty set.
What do I fail to understand?
combinatorics discrete-mathematics
edited Mar 21 at 11:31
YuiTo Cheng
2,3694937
$endgroup$
add a comment |
$begingroup$
Problem. How many subsets of ${1,2,3,ldots,100}$ contain all the even numbers?
I am not sure what is meant by "contain all the even numbers". If we are talking about all the even numbers between $1$ and $100$ (including $100$), then there is only one subset but if we are talking about all subsets whose elements are only even numbers, then since there are $50$ even numbers between $1$ and $100$ (including $100$), then there are $2^{50}-1$ subsets of the original set whose elements are only even numbers and the $-1$ is just for subtracting the empty set.
What do I fail to understand?
combinatorics discrete-mathematics
edited Mar 21 at 11:31
YuiTo Cheng
2,3694937
$endgroup$
2
$begingroup$
The empty set also contains only even numbers in the sense that for all $xin emptyset$, $x=2k$ for some integer $k$.
$endgroup$
– molarmass
Mar 21 at 11:36
3
$begingroup$
It sounds to me like what's being asked for is subsets that contain all of the even numbers and, possibly, some of the odd numbers from $1$ to $100$.
$endgroup$
– Barry Cipra
Mar 21 at 11:40
1
$begingroup$
This is one of reasons why you don't only write down the result but also an explanation/proof to show what you did and why.
$endgroup$
– Dirk
Mar 21 at 11:44
add a comment |
$begingroup$
Problem. How many subsets of ${1,2,3,ldots,100}$ contain all the even numbers?
I am not sure what is meant by "contain all the even numbers". If we are talking about all the even numbers between $1$ and $100$ (including $100$), then there is only one subset but if we are talking about all subsets whose elements are only even numbers, then since there are $50$ even numbers between $1$ and $100$ (including $100$), then there are $2^{50}-1$ subsets of the original set whose elements are only even numbers and the $-1$ is just for subtracting the empty set.
What do I fail to understand?
combinatorics discrete-mathematics
edited Mar 21 at 11:31
YuiTo Cheng
2,3694937
$endgroup$
Problem. How many subsets of ${1,2,3,ldots,100}$ contain all the even numbers?
I am not sure what is meant by "contain all the even numbers". If we are talking about all the even numbers between $1$ and $100$ (including $100$), then there is only one subset but if we are talking about all subsets whose elements are only even numbers, then since there are $50$ even numbers between $1$ and $100$ (including $100$), then there are $2^{50}-1$ subsets of the original set whose elements are only even numbers and the $-1$ is just for subtracting the empty set.
What do I fail to understand?
combinatorics discrete-mathematics
combinatorics discrete-mathematics
edited Mar 21 at 11:31
YuiTo Cheng
2,3694937
edited Mar 21 at 11:31
YuiTo Cheng
2,3694937
edited Mar 21 at 11:31
YuiTo Cheng
2,3694937
edited Mar 21 at 11:31
YuiTo Cheng
2,3694937
edited Mar 21 at 11:31
YuiTo Cheng
2,3694937
2,3694937
asked Mar 21 at 11:29
user656433
2
$begingroup$
The empty set also contains only even numbers in the sense that for all $xin emptyset$, $x=2k$ for some integer $k$.
$endgroup$
– molarmass
Mar 21 at 11:36
3
$begingroup$
It sounds to me like what's being asked for is subsets that contain all of the even numbers and, possibly, some of the odd numbers from $1$ to $100$.
$endgroup$
– Barry Cipra
Mar 21 at 11:40
1
$begingroup$
This is one of reasons why you don't only write down the result but also an explanation/proof to show what you did and why.
$endgroup$
– Dirk
Mar 21 at 11:44
add a comment |
2
$begingroup$
The empty set also contains only even numbers in the sense that for all $xin emptyset$, $x=2k$ for some integer $k$.
$endgroup$
– molarmass
Mar 21 at 11:36
3
$begingroup$
It sounds to me like what's being asked for is subsets that contain all of the even numbers and, possibly, some of the odd numbers from $1$ to $100$.
$endgroup$
– Barry Cipra
Mar 21 at 11:40
1
$begingroup$
This is one of reasons why you don't only write down the result but also an explanation/proof to show what you did and why.
$endgroup$
– Dirk
Mar 21 at 11:44
2
2
$begingroup$
The empty set also contains only even numbers in the sense that for all $xin emptyset$, $x=2k$ for some integer $k$.
$endgroup$
– molarmass
Mar 21 at 11:36
$begingroup$
The empty set also contains only even numbers in the sense that for all $xin emptyset$, $x=2k$ for some integer $k$.
$endgroup$
– molarmass
Mar 21 at 11:36
3
3
$begingroup$
It sounds to me like what's being asked for is subsets that contain all of the even numbers and, possibly, some of the odd numbers from $1$ to $100$.
$endgroup$
– Barry Cipra
Mar 21 at 11:40
$begingroup$
It sounds to me like what's being asked for is subsets that contain all of the even numbers and, possibly, some of the odd numbers from $1$ to $100$.
$endgroup$
– Barry Cipra
Mar 21 at 11:40
1
1
$begingroup$
This is one of reasons why you don't only write down the result but also an explanation/proof to show what you did and why.
$endgroup$
– Dirk
Mar 21 at 11:44
$begingroup$
This is one of reasons why you don't only write down the result but also an explanation/proof to show what you did and why.
$endgroup$
– Dirk
Mar 21 at 11:44
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
I understand it literally. They ask you to find how many subsets of ${1, 2, ..., 100}$ contains all even numbers from the given set, which are ${2, 4, ..., 100}$. So, you only need to find the number of different subsets you can form from the given odd numbers.
answered Mar 21 at 11:50
SlowLearnerSlowLearner
18513
$endgroup$
add a comment |
$begingroup$
"Contain all the even numbers" means exactly that. Example:
${2,4,6,...,100,1}$ contains all the even numbers and $1$.
answered Mar 21 at 11:40
MaxMax
29216
$endgroup$
$begingroup$
This will only reinforce the ambiguity in the OP's mind! "Contain all the even numbers" is not exactly "contains all the even numbers and $1$." You should state explicitly that your example is one of the subsets to be counted.
$endgroup$
– TonyK
Mar 21 at 11:57
$begingroup$
It should be sufficiently clear that the condition "contains all the even numbers" is fulfilled by "contains all the even numbers and $1$".
$endgroup$
– Max
Mar 21 at 12:05
$begingroup$
If somebody asks me, "What does XYZ mean?", it is not very helpful to answer "XYZ means exactly that."
$endgroup$
– TonyK
Mar 21 at 13:03
$begingroup$
"Means exactly that" is idiomatic to "literally", and thats what I wanted to say: "Read this condition literally." Also, I did provide an example to clarify the point.
$endgroup$
– Max
Mar 21 at 13:32
add a comment |
$begingroup$
It is ambiguous. I would favour the first interpretation. It should have said "contain all/only even numbers" without "the" to imply the second interpretation.
Anywho, there is more than one subset that contains all the even numbers from $2$ to $100$. In fact, any superset of ${2,4,6,...,100}$ satisfies the requirement. Since any superset of the above set is a union of it with a subset of ${1,3,5,...,99}$, we have $2^{50}$ such subsets.
Your answer keeping in mind the second interpretation is correct upto the debatable exclusion of the null set.
answered Mar 21 at 11:40
Shubham JohriShubham Johri
5,558818
$endgroup$
1
$begingroup$
It is not ambiguous at all. It means "How many subsets of ${1,2,ldots, 99,100}$ contain all the numbers ${2,4,ldots,98,100}$ (and possibly more numbers)?"
$endgroup$
– TonyK
Mar 21 at 11:54
$begingroup$
They have to be unique subsets.
$endgroup$
– SlowLearner
Mar 21 at 12:07
1
$begingroup$
@OgnjenMojovic ${1,3,5,...,99}$ has $2^{50}$ unique subsets.
$endgroup$
– Shubham Johri
Mar 21 at 12:11
$begingroup$
@ShubhamJohri : Yes, you're right. I swapped subsets and permutations.
$endgroup$
– SlowLearner
Mar 21 at 12:12
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I understand it literally. They ask you to find how many subsets of ${1, 2, ..., 100}$ contains all even numbers from the given set, which are ${2, 4, ..., 100}$. So, you only need to find the number of different subsets you can form from the given odd numbers.
answered Mar 21 at 11:50
SlowLearnerSlowLearner
18513
$endgroup$
add a comment |
$begingroup$
I understand it literally. They ask you to find how many subsets of ${1, 2, ..., 100}$ contains all even numbers from the given set, which are ${2, 4, ..., 100}$. So, you only need to find the number of different subsets you can form from the given odd numbers.
answered Mar 21 at 11:50
SlowLearnerSlowLearner
18513
$endgroup$
add a comment |
$begingroup$
I understand it literally. They ask you to find how many subsets of ${1, 2, ..., 100}$ contains all even numbers from the given set, which are ${2, 4, ..., 100}$. So, you only need to find the number of different subsets you can form from the given odd numbers.
answered Mar 21 at 11:50
SlowLearnerSlowLearner
18513
$endgroup$
I understand it literally. They ask you to find how many subsets of ${1, 2, ..., 100}$ contains all even numbers from the given set, which are ${2, 4, ..., 100}$. So, you only need to find the number of different subsets you can form from the given odd numbers.
answered Mar 21 at 11:50
SlowLearnerSlowLearner
18513
answered Mar 21 at 11:50
SlowLearnerSlowLearner
18513
answered Mar 21 at 11:50
SlowLearnerSlowLearner
18513
answered Mar 21 at 11:50
SlowLearnerSlowLearner
18513
18513
add a comment |
add a comment |
$begingroup$
"Contain all the even numbers" means exactly that. Example:
${2,4,6,...,100,1}$ contains all the even numbers and $1$.
answered Mar 21 at 11:40
MaxMax
29216
$endgroup$
$begingroup$
This will only reinforce the ambiguity in the OP's mind! "Contain all the even numbers" is not exactly "contains all the even numbers and $1$." You should state explicitly that your example is one of the subsets to be counted.
$endgroup$
– TonyK
Mar 21 at 11:57
$begingroup$
It should be sufficiently clear that the condition "contains all the even numbers" is fulfilled by "contains all the even numbers and $1$".
$endgroup$
– Max
Mar 21 at 12:05
$begingroup$
If somebody asks me, "What does XYZ mean?", it is not very helpful to answer "XYZ means exactly that."
$endgroup$
– TonyK
Mar 21 at 13:03
$begingroup$
"Means exactly that" is idiomatic to "literally", and thats what I wanted to say: "Read this condition literally." Also, I did provide an example to clarify the point.
$endgroup$
– Max
Mar 21 at 13:32
add a comment |
$begingroup$
"Contain all the even numbers" means exactly that. Example:
${2,4,6,...,100,1}$ contains all the even numbers and $1$.
answered Mar 21 at 11:40
MaxMax
29216
$endgroup$
$begingroup$
This will only reinforce the ambiguity in the OP's mind! "Contain all the even numbers" is not exactly "contains all the even numbers and $1$." You should state explicitly that your example is one of the subsets to be counted.
$endgroup$
– TonyK
Mar 21 at 11:57
$begingroup$
It should be sufficiently clear that the condition "contains all the even numbers" is fulfilled by "contains all the even numbers and $1$".
$endgroup$
– Max
Mar 21 at 12:05
$begingroup$
If somebody asks me, "What does XYZ mean?", it is not very helpful to answer "XYZ means exactly that."
$endgroup$
– TonyK
Mar 21 at 13:03
$begingroup$
"Means exactly that" is idiomatic to "literally", and thats what I wanted to say: "Read this condition literally." Also, I did provide an example to clarify the point.
$endgroup$
– Max
Mar 21 at 13:32
add a comment |
$begingroup$
"Contain all the even numbers" means exactly that. Example:
${2,4,6,...,100,1}$ contains all the even numbers and $1$.
answered Mar 21 at 11:40
MaxMax
29216
$endgroup$
"Contain all the even numbers" means exactly that. Example:
${2,4,6,...,100,1}$ contains all the even numbers and $1$.
answered Mar 21 at 11:40
MaxMax
29216
answered Mar 21 at 11:40
MaxMax
29216
answered Mar 21 at 11:40
MaxMax
29216
answered Mar 21 at 11:40
MaxMax
29216
29216
$begingroup$
This will only reinforce the ambiguity in the OP's mind! "Contain all the even numbers" is not exactly "contains all the even numbers and $1$." You should state explicitly that your example is one of the subsets to be counted.
$endgroup$
– TonyK
Mar 21 at 11:57
$begingroup$
It should be sufficiently clear that the condition "contains all the even numbers" is fulfilled by "contains all the even numbers and $1$".
$endgroup$
– Max
Mar 21 at 12:05
$begingroup$
If somebody asks me, "What does XYZ mean?", it is not very helpful to answer "XYZ means exactly that."
$endgroup$
– TonyK
Mar 21 at 13:03
$begingroup$
"Means exactly that" is idiomatic to "literally", and thats what I wanted to say: "Read this condition literally." Also, I did provide an example to clarify the point.
$endgroup$
– Max
Mar 21 at 13:32
add a comment |
$begingroup$
This will only reinforce the ambiguity in the OP's mind! "Contain all the even numbers" is not exactly "contains all the even numbers and $1$." You should state explicitly that your example is one of the subsets to be counted.
$endgroup$
– TonyK
Mar 21 at 11:57
$begingroup$
It should be sufficiently clear that the condition "contains all the even numbers" is fulfilled by "contains all the even numbers and $1$".
$endgroup$
– Max
Mar 21 at 12:05
$begingroup$
If somebody asks me, "What does XYZ mean?", it is not very helpful to answer "XYZ means exactly that."
$endgroup$
– TonyK
Mar 21 at 13:03
$begingroup$
"Means exactly that" is idiomatic to "literally", and thats what I wanted to say: "Read this condition literally." Also, I did provide an example to clarify the point.
$endgroup$
– Max
Mar 21 at 13:32
$begingroup$
This will only reinforce the ambiguity in the OP's mind! "Contain all the even numbers" is not exactly "contains all the even numbers and $1$." You should state explicitly that your example is one of the subsets to be counted.
$endgroup$
– TonyK
Mar 21 at 11:57
$begingroup$
This will only reinforce the ambiguity in the OP's mind! "Contain all the even numbers" is not exactly "contains all the even numbers and $1$." You should state explicitly that your example is one of the subsets to be counted.
$endgroup$
– TonyK
Mar 21 at 11:57
$begingroup$
It should be sufficiently clear that the condition "contains all the even numbers" is fulfilled by "contains all the even numbers and $1$".
$endgroup$
– Max
Mar 21 at 12:05
$begingroup$
It should be sufficiently clear that the condition "contains all the even numbers" is fulfilled by "contains all the even numbers and $1$".
$endgroup$
– Max
Mar 21 at 12:05
$begingroup$
If somebody asks me, "What does XYZ mean?", it is not very helpful to answer "XYZ means exactly that."
$endgroup$
– TonyK
Mar 21 at 13:03
$begingroup$
If somebody asks me, "What does XYZ mean?", it is not very helpful to answer "XYZ means exactly that."
$endgroup$
– TonyK
Mar 21 at 13:03
$begingroup$
"Means exactly that" is idiomatic to "literally", and thats what I wanted to say: "Read this condition literally." Also, I did provide an example to clarify the point.
$endgroup$
– Max
Mar 21 at 13:32
$begingroup$
"Means exactly that" is idiomatic to "literally", and thats what I wanted to say: "Read this condition literally." Also, I did provide an example to clarify the point.
$endgroup$
– Max
Mar 21 at 13:32
add a comment |
$begingroup$
It is ambiguous. I would favour the first interpretation. It should have said "contain all/only even numbers" without "the" to imply the second interpretation.
Anywho, there is more than one subset that contains all the even numbers from $2$ to $100$. In fact, any superset of ${2,4,6,...,100}$ satisfies the requirement. Since any superset of the above set is a union of it with a subset of ${1,3,5,...,99}$, we have $2^{50}$ such subsets.
Your answer keeping in mind the second interpretation is correct upto the debatable exclusion of the null set.
answered Mar 21 at 11:40
Shubham JohriShubham Johri
5,558818
$endgroup$
1
$begingroup$
It is not ambiguous at all. It means "How many subsets of ${1,2,ldots, 99,100}$ contain all the numbers ${2,4,ldots,98,100}$ (and possibly more numbers)?"
$endgroup$
– TonyK
Mar 21 at 11:54
$begingroup$
They have to be unique subsets.
$endgroup$
– SlowLearner
Mar 21 at 12:07
1
$begingroup$
@OgnjenMojovic ${1,3,5,...,99}$ has $2^{50}$ unique subsets.
$endgroup$
– Shubham Johri
Mar 21 at 12:11
$begingroup$
@ShubhamJohri : Yes, you're right. I swapped subsets and permutations.
$endgroup$
– SlowLearner
Mar 21 at 12:12
add a comment |
$begingroup$
It is ambiguous. I would favour the first interpretation. It should have said "contain all/only even numbers" without "the" to imply the second interpretation.
Anywho, there is more than one subset that contains all the even numbers from $2$ to $100$. In fact, any superset of ${2,4,6,...,100}$ satisfies the requirement. Since any superset of the above set is a union of it with a subset of ${1,3,5,...,99}$, we have $2^{50}$ such subsets.
Your answer keeping in mind the second interpretation is correct upto the debatable exclusion of the null set.
answered Mar 21 at 11:40
Shubham JohriShubham Johri
5,558818
$endgroup$
1
$begingroup$
It is not ambiguous at all. It means "How many subsets of ${1,2,ldots, 99,100}$ contain all the numbers ${2,4,ldots,98,100}$ (and possibly more numbers)?"
$endgroup$
– TonyK
Mar 21 at 11:54
$begingroup$
They have to be unique subsets.
$endgroup$
– SlowLearner
Mar 21 at 12:07
1
$begingroup$
@OgnjenMojovic ${1,3,5,...,99}$ has $2^{50}$ unique subsets.
$endgroup$
– Shubham Johri
Mar 21 at 12:11
$begingroup$
@ShubhamJohri : Yes, you're right. I swapped subsets and permutations.
$endgroup$
– SlowLearner
Mar 21 at 12:12
add a comment |
$begingroup$
It is ambiguous. I would favour the first interpretation. It should have said "contain all/only even numbers" without "the" to imply the second interpretation.
Anywho, there is more than one subset that contains all the even numbers from $2$ to $100$. In fact, any superset of ${2,4,6,...,100}$ satisfies the requirement. Since any superset of the above set is a union of it with a subset of ${1,3,5,...,99}$, we have $2^{50}$ such subsets.
Your answer keeping in mind the second interpretation is correct upto the debatable exclusion of the null set.
answered Mar 21 at 11:40
Shubham JohriShubham Johri
5,558818
$endgroup$
It is ambiguous. I would favour the first interpretation. It should have said "contain all/only even numbers" without "the" to imply the second interpretation.
Anywho, there is more than one subset that contains all the even numbers from $2$ to $100$. In fact, any superset of ${2,4,6,...,100}$ satisfies the requirement. Since any superset of the above set is a union of it with a subset of ${1,3,5,...,99}$, we have $2^{50}$ such subsets.
Your answer keeping in mind the second interpretation is correct upto the debatable exclusion of the null set.
answered Mar 21 at 11:40
Shubham JohriShubham Johri
5,558818
answered Mar 21 at 11:40
Shubham JohriShubham Johri
5,558818
answered Mar 21 at 11:40
Shubham JohriShubham Johri
5,558818
answered Mar 21 at 11:40
Shubham JohriShubham Johri
5,558818
5,558818
1
$begingroup$
It is not ambiguous at all. It means "How many subsets of ${1,2,ldots, 99,100}$ contain all the numbers ${2,4,ldots,98,100}$ (and possibly more numbers)?"
$endgroup$
– TonyK
Mar 21 at 11:54
$begingroup$
They have to be unique subsets.
$endgroup$
– SlowLearner
Mar 21 at 12:07
1
$begingroup$
@OgnjenMojovic ${1,3,5,...,99}$ has $2^{50}$ unique subsets.
$endgroup$
– Shubham Johri
Mar 21 at 12:11
$begingroup$
@ShubhamJohri : Yes, you're right. I swapped subsets and permutations.
$endgroup$
– SlowLearner
Mar 21 at 12:12
add a comment |
1
$begingroup$
It is not ambiguous at all. It means "How many subsets of ${1,2,ldots, 99,100}$ contain all the numbers ${2,4,ldots,98,100}$ (and possibly more numbers)?"
$endgroup$
– TonyK
Mar 21 at 11:54
$begingroup$
They have to be unique subsets.
$endgroup$
– SlowLearner
Mar 21 at 12:07
1
$begingroup$
@OgnjenMojovic ${1,3,5,...,99}$ has $2^{50}$ unique subsets.
$endgroup$
– Shubham Johri
Mar 21 at 12:11
$begingroup$
@ShubhamJohri : Yes, you're right. I swapped subsets and permutations.
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– SlowLearner
Mar 21 at 12:12
1
1
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It is not ambiguous at all. It means "How many subsets of ${1,2,ldots, 99,100}$ contain all the numbers ${2,4,ldots,98,100}$ (and possibly more numbers)?"
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– TonyK
Mar 21 at 11:54
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It is not ambiguous at all. It means "How many subsets of ${1,2,ldots, 99,100}$ contain all the numbers ${2,4,ldots,98,100}$ (and possibly more numbers)?"
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– TonyK
Mar 21 at 11:54
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They have to be unique subsets.
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– SlowLearner
Mar 21 at 12:07
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They have to be unique subsets.
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– SlowLearner
Mar 21 at 12:07
1
1
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@OgnjenMojovic ${1,3,5,...,99}$ has $2^{50}$ unique subsets.
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– Shubham Johri
Mar 21 at 12:11
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@OgnjenMojovic ${1,3,5,...,99}$ has $2^{50}$ unique subsets.
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– Shubham Johri
Mar 21 at 12:11
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@ShubhamJohri : Yes, you're right. I swapped subsets and permutations.
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– SlowLearner
Mar 21 at 12:12
$begingroup$
@ShubhamJohri : Yes, you're right. I swapped subsets and permutations.
$endgroup$
– SlowLearner
Mar 21 at 12:12
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The empty set also contains only even numbers in the sense that for all $xin emptyset$, $x=2k$ for some integer $k$.
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– molarmass
Mar 21 at 11:36
3
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It sounds to me like what's being asked for is subsets that contain all of the even numbers and, possibly, some of the odd numbers from $1$ to $100$.
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– Barry Cipra
Mar 21 at 11:40
1
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This is one of reasons why you don't only write down the result but also an explanation/proof to show what you did and why.
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– Dirk
Mar 21 at 11:44