First, it should be clear to you that $k \le \left\lceil {{n \over 2}} \right\rceil $ (that is the ceiling function).

Think of a bit-string consisting of k ones and n-k zeros.

Any rearrangement of that string represents a k-element subset of an n-set.

How many rearrangements are there in which no two ones are consecutive?

Well the zeros create n-k+1 places to place the ones, chose k of them.