# How many subsets of $\{1,2,3,\ldots,100\}$ contain all the even numbers?

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?

• The empty set also contains only even numbers in the sense that for all $x\in \emptyset$, $x=2k$ for some integer $k$. – molarmass Mar 21 at 11:36
• 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 $100$. – Barry Cipra Mar 21 at 11:40
• 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. – Dirk Mar 21 at 11:44

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.

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

• 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. – TonyK Mar 21 at 11:57
• It should be sufficiently clear that the condition "contains all the even numbers" is fulfilled by "contains all the even numbers and $1$". – Max Mar 21 at 12:05
• If somebody asks me, "What does XYZ mean?", it is not very helpful to answer "XYZ means exactly that." – TonyK Mar 21 at 13:03
• "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. – Max Mar 21 at 13:32

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.

• 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)?" – TonyK Mar 21 at 11:54
• They have to be unique subsets. – MathsLearner Mar 21 at 12:07
• @OgnjenMojovic $\{1,3,5,...,99\}$ has $2^{50}$ unique subsets. – Shubham Johri Mar 21 at 12:11
• @ShubhamJohri : Yes, you're right. I swapped subsets and permutations. – MathsLearner Mar 21 at 12:12