The Factorial Pyramid

\[\dfrac{\dfrac{ \; \;\; \;\; \;\; \; 100! \; \;\; \;\; \;\; \;}{\dfrac{\; \;\; \;\; \;99!\; \;\; \;\; \;}{\dfrac{ \;\; \;\; \;98!\; \;\; \; }{\dfrac { \;\; \;\vdots\; \; \;}{\dfrac{ \; \;2!\; \; }{\dfrac{ \; 1! \; }{ 0! } }} }}}}{2^{50}} \]

If the expression above can be simplified to \(n!\), find \(n\).

Notation: \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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