Thanks in advance!
Let be a balanced subset of . Then the number of elements of must be even. Furthermore, can be either . Assume we wish to find all balanced subsets with elements, . To do this we need to choose even integers, and and odd integers. There are ways to choose an even integer and to choose odd integers. In total there are ways to create a balanced subset with integers. This sum is taken over . And thus, there are a total of balanced subsets.
Now a continuation of this question, I have came up with the formula
Prove combinatorially that this is true for all integers n>=1.
I have this so far:
1. Choose integers from the
2. Choose all the even integers in that set
3. Choose the odd that are not in that set
How can I turn this into a more formal proof?