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Bijective counting

  1. Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of \(S\) are there that satisfy \(A \subseteq B\) ?

  2. Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of \(S\) are there such that \(A\) and \(B\) are disjoint ?

  3. Consider a convex polygon on \(n\) points such that no three points are collinear. A chord is a line segment between two non-adjacent vertices. If all chords are drawn, in how many interior points do they intersect ? Assume no three chords intersect at the same point.

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