Counting Subsets

In the setting of Finite Combinatorics: Notation, let $A$ be a finite set with $\#A=n$, and let $k\in\mathbb{N}_{0}$.

(i) Subsets of a given size. $\mathcal{P}_{k}(A)$ is finite, and

$$\#\mathcal{P}_{k}(A)=\binom{n}{k}.$$

(ii) All subsets. $\mathcal{P}(A)$ is finite, and

$$\#\mathcal{P}(A)=2^{n}.$$

(iii) Grouping by size. For every map $g:\mathbb{N}_{0}\to\mathbb{R}$,

$$\sum_{S\in\mathcal{P}(A)}g(\#S)=\sum_{j=0}^{n}\binom{n}{j}g(j).$$