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The coefficient of $x^{91}$ in the series $^{100}{C_1}\,{2^8}.\,{\left( {1\, - \,x} \right)^{99}}\, + {\,^{100}}{C_2}\,{2^7}.\,{\left( {1\, - \,x} \right)^{98}}\, + {\,^{100}}{C_3}\,{2^6}.\,{\left( {1\, - \,x} \right)^{97}}\, + \,....\, + {\,^{100}}{C_9}\,{\left( {1\, - \,x} \right)^{91}}$ is equal to -
$^{100}{C_{10}}({2^9})$
$^{100}{C_{10}}({2^9 - 3^9})$
$^{100}{C_{9}}({2^9 - 3^9})$
$^{100}{C_{9}}({3^9})$
Solution
coefficient of
${{\rm{x}}^{91}} = – {{\rm{(}}^{100}}{{\rm{C}}_1} \cdot {2^8}{{\rm{.}}^{99}}{{\rm{C}}_{91}} + \ldots . + {\,^{100}}{{\rm{C}}_9} \cdot {2^0}{{\rm{.}}^{91}}{{\rm{C}}_{91}})$
$ = – \sum\limits_{r = 0}^8 {^{100}{C_{r + 1}}{{.2}^{\left( {8 – r} \right)}}{.^{99 – r}}{C_{91}}} $
$ = – {2^9}{ \cdot ^{100}}{{\rm{C}}_9}\sum\limits_{r = 0}^8 {{\,^9}{C_{r + 1}}} {\left( {\frac{1}{2}} \right)^{r + 1}}$
$ = {\,^{100}}{{\rm{C}}_9}\left( {{2^9} – {3^9}} \right)$
