# Combinatorics and sets 2

1. Dec 2, 2016

### agargento

1. The problem statement, all variables and given/known data

Given {1,2,...,n}, n is an even number. What are all the possible subsets that contain only even numbers? (Notice that ∅ is also defined as such a subset).

2. Relevant equations

2n - all possibilities for group A with n objects

3. The attempt at a solution

I think the answer is 2n/2 .... to include just even numbers. Is my reasoning correct?

2. Dec 2, 2016

### FactChecker

You should add a few sentences to explain your reasoning. And the question is "what are", not "how many", so can you say something more about them?.

3. Dec 3, 2016

### agargento

Just bad translation on my part. "How many" is more correct. As for reasoning, we were taught in class that for n objects, you have 2n possibilities for subsets. So I thought that because we have n/2 objects (we want only even numbers) 2n/2 is the answer.

4. Dec 3, 2016

### FactChecker

That is better. It never hurts to add a brief explanation like that.

5. Dec 3, 2016

### agargento

But is it correct?

6. Dec 3, 2016

### PeroK

It is correct. The odd numbers are irrelevant.

7. Dec 3, 2016

### agargento

Hmm ok. But what about ∅ ? If it is even, it should be included, but it does not seem to be included in {1...n}...

8. Dec 3, 2016

### PeroK

That's the empty set. That is included as one of the $2^n$ subsets. If you are looking for non-empty subsets then there are only $2^n -1$ of those.

In this case you were explicitly told to count the empty set.

9. Dec 3, 2016

### agargento

Oh now I got it. Thanks!