Selecting distinct $A,B,C⊆\{1,2,3,4\}$ such that $A\cup B\cup C=S$

Question

Let $S$ be the set $\{1,2,3,4\}$. Now how many ways are there to select three distinct subsets $A$, $B$ and $C$ such that $A\cup B\cup C=S$?

I am confused how to approach this question. Only thing that came to my mind is to make cases but it's gonna take so much time. Is there another way of doing it?

Answer 1

For each $x\in S$, you have seven choices.

So the required number of ways is $7^4=2401$

\begin{array}{|c|c|c|c|} \hline \hline &\color{magenta}A &\color{magenta}B&\color{magenta}C\\ \hline \color{magenta}1&\checkmark &\checkmark&\checkmark\\ \color{magenta}2&\times &\checkmark&\checkmark\\ \color{magenta}3&\checkmark &\times&\checkmark\\ \color{magenta}4&\checkmark &\checkmark&\times\\ \color{magenta}5&\times &\times&\checkmark\\ \color{magenta}6&\checkmark &\times&\times\\ \color{magenta}7&\times &\checkmark&\times\\ \end{array}