Let \(S\,\) be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of \(S\,\) so that the union of the two subsets is \(S\,\)? The order of selection does not matter; for example, the pair of subsets \(\{a, c\}\,\), \(\{b, c, d, e, f\}\,\) represents the same selection as the pair \(\{b, c, d, e, f\}\,\), \(\{a, c\}\,\).

(第十一届AIME1993 第8题)