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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?

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    $\begingroup$ The empty set also contains only even numbers in the sense that for all $x\in \emptyset$, $x=2k$ for some integer $k$. $\endgroup$ – molarmass Mar 21 at 11:36
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    $\begingroup$ 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
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    $\begingroup$ 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
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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.

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"Contain all the even numbers" means exactly that. Example: $\{2,4,6,...,100,1\}$ contains all the even numbers and $1$.

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  • $\begingroup$ 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
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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.

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    $\begingroup$ 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. $\endgroup$ – MathsLearner Mar 21 at 12:07
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    $\begingroup$ @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. $\endgroup$ – MathsLearner Mar 21 at 12:12

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