Official solution
\[f\left( x \right) = {x^n} \Rightarrow f\left( 1 \right) = 1\]
\[f'\left( x \right) = n{x^{n - 1}} \Rightarrow f'\left( 1 \right) = n\]
\[f''\left( x \right) = n\left( {n - 1} \right){x^{n - 2}}\]
\[ \Rightarrow f''\left( 1 \right) = n\left( {n - 1} \right)\]
\[\therefore\] \[{f^n}\left( x \right) = n!\]
\[ \Rightarrow {f^n}\left( 1 \right) = n!\]
\[ = 1 - {n \over {1!}} + {{n\left( {n - 1} \right)} \over {2!}}{{n\left( {n - 1} \right)\left( {n - 2} \right)} \over {3!}}\]
\[\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + .... + {\left( { - 1} \right)^n}{{n!} \over {n!}}\]
\[ = {}^n\,{C_0} - {}^n\,{C_1} + {}^n\,{C_2} - {}^n\,{C_3}\]
\[\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + ...... + {\left( { - 1} \right)^n}\,{}^n{C_n} = 0\]