I am self-studying Discrete Mathematics, and I have the following question:
Let $X$ be a set with exactly $n$ distinct elements. How many elements does have $\mathcal{P}(X)?$
I know that there are $2^{n}$ subsets. I know how to prove it by using induction on $n$, but I want to solve this question by counting all the elements in $\mathcal{P}(X).$ I am not allowed to use the binomial theorem, or binomial coefficients because they were not defined yet. Is that possible?