I know a couple of ways to do this. However, I couldn't quite follow the logic of the accepted answer in this question…
What is the proof that the total number of subsets of a set is $2^n$?
…which says that there are two possibilities of an element being present in any subset of a given set, so all of them are multiplied.
I fail to understand exactly how many times I should do that. And that exactly is the answer. It doesn't seem that the answer explains this issue, but how does it work then?
Best Answer
You should multiply $2$ by itself $n$ times, where $n$ is the number of elements in your set. Consider the following:
You have a set $S$ with $n$ objects, and there are $k$ possible subsets you can make. Now add another object to $S$, that you have $n+1$ objects. How many subsets will you have now? Well you can for sure have at least as many as before, namely $k$. But you can now just add the $n+1^{th}$ term to each of the $k$ subsets, so you create another $k$ subsets. So now the number of possible subsets you can have is $2k$.
Moral: Adding an element to a set doubles the number of possible subsets.
Start with a set of $1$ object, and notice that you have $2$ possible subsets. By the above, if we add an element to the set, we have $2 \cdot 2=4$ possible subsets, if we add another element we have $4 \cdot 2$ possible subsets, and in general if we have $n$ elements we will have $2^n$ possible subsets.
Bonus: The basic idea of what we did above is called mathematical induction.