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

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?

• It would be helpful if you clarified a couple of things? Can $A,B$, or $C$ be empty? Is the order of $A,B$, and $C$ important? For instance does $A=\{1\}$, $B=\{2\}$, $C=\{3,4\}$ count as the same or different than $A=\{2\}$, $B=\{1\}$, $C=\{3,4\}$. Different answers to these questions will change the count of possibilities. Sep 24, 2021 at 18:52

For each $$x\in S$$, you have seven choices.
So the required number of ways is $$7^4=2401$$
$$\begin{array}{|c|c|c|c|} \hline \hline &\color{magenta}A &\color{magenta}B&\color{magenta}C\\ \hline \color{magenta}1&\checkmark &\checkmark&\checkmark\\ \color{magenta}2&\times &\checkmark&\checkmark\\ \color{magenta}3&\checkmark &\times&\checkmark\\ \color{magenta}4&\checkmark &\checkmark&\times\\ \color{magenta}5&\times &\times&\checkmark\\ \color{magenta}6&\checkmark &\times&\times\\ \color{magenta}7&\times &\checkmark&\times\\ \end{array}$$
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