• # question_answer If n is an integer greater than 1, then $a-{{\,}^{n}}{{C}_{1}}(a-1)+{{\,}^{n}}{{C}_{2}}(a-2)-....+{{(-1)}^{n}}(a-n)$ is equal to: A) $a$                                      B)                  $0$                                      C)                  ${{a}^{2}}$                                       D)                  ${{2}^{n}}$

$a-{{\,}^{n}}{{C}_{1}}(a-1)+{{\,}^{n}}{{C}_{2}}(a-2)-$ $....+{{(-1)}^{n}}(a-n)$                 $=a({{\,}^{n}}{{C}_{0}}-{{\,}^{n}}{{C}_{1}}{{+}^{n}}{{C}_{2}}{{-}^{n}}{{C}_{3}}+....+{{(-1)}^{n}}\,{{\,}^{n}}{{C}_{n}})$                 $+\,({{\,}^{n}}{{C}_{1}}-2{{\,}^{n}}{{C}_{2}}+3{{\,}^{n}}{{C}_{3}}-....+{{(-1)}^{n+1}}n{{\,}^{n}}{{C}_{n}})$                                                                                 ?(i) As we know ${{(1-x)}^{n}}={{\,}^{n}}{{C}_{0}}-{{x}^{n}}{{C}_{1}}+{{x}^{2n}}{{C}_{2}}-{{x}^{3}}{{\,}^{n}}C{{\,}_{3}}+$ $....+{{(-1)}^{n}}{{x}^{n}}{{\,}^{n}}{{C}_{n}}$                    ?(ii) Put $x=1,$we get $0={{\,}^{n}}{{C}_{0}}={{\,}^{n}}{{C}_{1}}{{+}^{n}}{{C}_{2}}{{-}^{n}}{{C}_{3}}+...+{{(-1)}^{n}}{{C}_{n}}$ On differentiating Eq. (ii) w.r.t.$x,$ we get $n{{(1-x)}^{n-1}}=-{{\,}^{n}}{{C}_{1}}+2x{{\,}^{n}}{{C}_{2}}-3{{x}^{2}}{{\,}^{n}}{{C}_{3}}+...$ $+{{(-1)}^{n}}n\,{{x}^{n-1}}{{\,}^{n}}{{C}_{n}}$                 Put $x=1$                 $0=-{{\,}^{n}}{{C}_{1}}+2{{\,}^{n}}{{C}_{2}}-3\,{{\,}^{n}}{{C}_{3}}+....+{{(-1)}^{n-1}}n{{\,}^{n}}{{C}_{n}}$                 From Eq. (i)                 $a-{{\,}^{n}}{{C}_{1}}(a-1)+{{\,}^{n}}{{C}_{2}}(a-2)-...+{{(-1)}^{n}}(a-n)$ $=a(0)+0=0$