The coefficient of x^{91} in the series ^{100 | vedclass

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Question

coefficient of

${{\rm{x}}^{91}} =  - {{\rm{(}}^{100}}{{\rm{C}}_1} \cdot {2^8}{{\rm{.}}^{99}}{{\rm{C}}_{91}} +  \ldots . + {\,^{100}}{{\rm

Vedclass · Accepted Answer

$^{100}{C_{9}}({2^9 - 3^9})$

Vedclass · Answer

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)$

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