If $$f\left( x \right) = {x^n},$$ then the value of
$$f\left( 1 \right) - {{f'\left( 1 \right)} \over {1!}} + {{f''\left( 1 \right)} \over {2!}} - {{f'''\left( 1 \right)} \over {3!}} + ..........{{{{\left( { - 1} \right)}^n}{f^n}\left( 1 \right)} \over {n!}}$$ is
CHECK ANSWER
Explanation $$f\left( x \right) = {x^n} \Rightarrow f\left( 1 \right) = 1$$
$$f'\left( x \right) = n{x^{n - 1}} \Rightarrow f'\left( 1 \right) = n$$
$$f''\left( x \right) = n\left( {n - 1} \right){x^{n - 2}}$$
$$ \Rightarrow f''\left( 1 \right) = n\left( {n - 1} \right)$$
$$\therefore$$ $${f^n}\left( x \right) = n!$$
$$ \Rightarrow {f^n}\left( 1 \right) = n!$$
$$ = 1 - {n \over {1!}} + {{n\left( {n - 1} \right)} \over {2!}}{{n\left( {n - 1} \right)\left( {n - 2} \right)} \over {3!}}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + .... + {\left( { - 1} \right)^n}{{n!} \over {n!}}$$
$$ = {}^n\,{C_0} - {}^n\,{C_1} + {}^n\,{C_2} - {}^n\,{C_3}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + ...... + {\left( { - 1} \right)^n}\,{}^n{C_n} = 0$$