Question

How many subsets containing an even number of elements are there in a set of 97 objects?

Answer 1

You get extract $2i$ objects out of 97 object in this number of ways:

$\displaystyle \binom{97}{2i} = \dfrac{97!}{(2i)!(97-2i)!}$

So, the number of all possible subsets is

$\displaystyle \binom{97}{0} + \binom{97}{2} + \ldots + \binom{97}{96} = \sum_{i=0}^{48}\binom{97}{2i} = 79228162514264337593543950336$