combinatorics - Prove that $P(X)$ has exactly $binom nk$ subsets of $X$ of $k$ elements each.

Let set $X$ consist of $n$ members. $P(X)$ is power set of $X$.

Prove that set $P(X)$ has exactly $$\binom nk = \frac{n!}{k!(n-k)!}$$ subsets of $X$ of $k$ elements each. Hence, show that $P(X)$ contains $2^n$ members.
Hint: use the binomial expansion of $(1 + 1)^n$.

I'm trying to get more into math (proof writing first) so I got this from a book about Real Analysis. It's a short first sub-chapter which was an intro review of set theory and I have absolutely no idea how to do this particular problem given what has been said in 8 pages.

I 'know' that it's true, since I've tried it. I can also somewhat prove that $P(X)$ has $2^n$ elements thinking of binary numbers. I can also see that the binomial expansion of $(1 + 1)^n$ does tell the number of subsets of $X$ that has $k$ elements via the coefficients. I just don't know how they all fit together given that the author thinks they should.

I've also seen some questions in here proving the summation of $\binom nk = 2^n$, which was suppose to help, but I couldn't understand them at all given my limited math/combinatorics knowledge.