TITLE: Binomial Theorem past exam question, what do I do?
QUESTION [6 upvotes]: I have trouble understanding what I'm supposed to do in some of these math questions. 
Here's an exam question from an old exam:
Let $A$ be a set with $n$ elements. The number of subsets of $A$ with $k$ elements is ${n \choose k}$, or:
$$ C(n,k) = \frac{n!}{k!(n - k)!} $$
Show that there are as many subsets having an odd number of elements as there are subsets with an even number of elements.
HINT: use the Binomial Theorem in the form:
$$ (1 + x)^n = \sum_{k = 0}^n {n \choose k} x^k$$
Then set $x = -1$.
Do I just choose an arbitrary number for $n$ then set $x = -1$ and work it out?
I don't know why I have such a problem understanding these questions. Thanks for your help.

REPLY [3 votes]: Hint: Don't use the hint. 
We assume (necessarily) that $n\ge 1$ and let $a$ be an element of $A$.
There is an obvious bijection between a) the odd sets containing $a$ and the even sets not containing $a$ (drop $a$ from the set) and b) the odd sets not containing $a$ and the even sets containing $a$ (add $a$ to the set).
Together this gives a bijection of odd and even sets, especially the numbers are the same.