# A subset of a subset of a subset of a subset of a set of $\{1,2,\cdots,10\}$

• There is a set of $$10$$ first natural numbers, $$S = \{1,2,\cdots,10\}$$.
• Alice picks a subset of it, say $$A \subseteq S$$.
• Bob picks a subset of it, say $$B \subseteq A$$.
• Charlie picks a subset of it, say $$C \subseteq B$$.
• Dave picks a subset of it, say $$D \subseteq C$$.

How many different ways of their picking are there?
i.e. How many different tuples of $$\langle A,B,C,D \rangle$$ are there?

• Subsets, or proper subsets? (i.e. is the null set valid? Is the full set valid?) – Chris Cudmore Oct 18 '19 at 13:27
• is this a math question rather than a puzzle??? – Omega Krypton Oct 18 '19 at 13:28
• They are subsets, i.e. it is possible for $A = S$ and also $A = \{\}$. Idk if this is a good puzzle here because I guess the way to solve this is pretty fun as a math puzzle. Tho it's perfectly fine if it's considered off-topic if people say so.. ^^ – athin Oct 18 '19 at 13:31
• sorry i have to -1 this as this is off-topic... – Omega Krypton Oct 18 '19 at 13:45
• Seeing the answer, I would consider this a maths puzzle as there is a clever and quick way to solve it which is not immediately obvious. – hexomino Oct 18 '19 at 13:51

For each element we can choose the last person to have it in their set. There are $$5$$ possibilities for each element, namely None, Alice, Bob, Charlie, or Dave. Each element is independent of all the others. Therefore there are $$5^{10}=9765625$$ ways to arrange the elements amongst the sets.