Interpretation of $\sum^{n}_{k=0}\limits{\binom{n}{k}=2^{n}}$

Question

By using methods of combinatorics, one can prove that $\sum^{n}_{k=0}\limits{\binom{n}{k}=2^{n}}$.

However, does this have formula have a practical meaning? The binomial coefficient $\binom{n}{k}$ can be interpreted as the number of possibilities to choose k elements from a set of n elements, if the order is not important. However, can this definition be somehow extended to this formula?

Thank you in advance.

Answer 1

Say you want to create a subset of any size from $n$ elements.

$\binom{n}{0}$ creates all subsets of size $0$, $\cdots, \binom{n}{i}$ creates all subsets of size $i, \cdots, \binom{n}{n}$ creates all subsets of size $n$.

So $\sum_{k=0}^n \binom{n}{k}$ counts all subsets from $n$ elements of any size.

However, we can also do this by considering if each element is in the set or not in the set. So there are $2$ options for each of the $n$ elements, or $2^n$ total subsets.

Answer 2

Hint: What's the total number of subsets of set of $n$ elements?