Problem
COM-B1-M04-P008 Sum of Subset Sizes
#8
★★☆☆☆ Level 2 of 5
For a set of \(5\) elements, find the sum of the sizes of all its subsets.
Count pairs \((S,x)\), where \(x\in S\).
Each of the \(5\) elements belongs to exactly half of all subsets, i.e. \(2^4=16\) subsets. The sum of subset sizes equals the number of pairs \((S,x)\), so it is \(5\cdot16=80\).
Concrete version of the identity.