Given the set $A=\{1,2,3,4,5,6\}$, find:
(1) The number of all subsets of $A$;
(2) The number of all non-empty proper subsets of $A$;
(3) The number of all non-empty subsets of $A$;
(4) The sum of all elements in all non-empty subsets of $A$.
(1) The number of all subsets of $A$ is $2^{6}=64$.
(2) The number of all non-empty proper subsets of $A$ is $2^{6}-2=62$.
(3) The number of all non-empty subsets of $A$ is $2^{6}-1=63$.
(4) The sum of all elements in all non-empty subsets of $A$ is $(1+2+3+4+5+6) \cdot 2^{6-1}=672$.

Key Points for Solving Generally, when a set $A$ has $n\left(n \in \mathbf{N}^{*}\right)$ elements, the number of its subsets is $2^{n}$; the number of non-empty subsets and proper subsets is $2^{n}-1$; the number of non-empty proper subsets is $2^{n}-2$; when finding the sum of all elements in all non-empty subsets, the key is to determine how many times each element in set $A$ appears in its subsets (i.e., how many times it is added), which is the number of subsets of the set formed by the remaining elements except for that element.