How many subsets are there of a set of cardinality $n$?
The problem can be solved in several ways.

We can simply content ourselves with linking it to sequences of 0s and 1s of length $n$ (there are $2^{n}$ of them) which are a form of representation of a subset, since we can, for example, say that the subset associated with a sequence is constructed such that the $i^{\text{th}}$ element is in the subset if and only if the $i^{\text{th}}$ term of the sequence is 1. For each sequence, there is therefore a set, and for each set, there is a unique sequence. There are therefore exactly $2^{n}$ subsets of a set of cardinality $n$.

Another method of proof involves reasoning by induction on $n$. We leave it to the reader to write the proof, which should not present any difficulty.