Coefficient of {x^{53}} in following expansion Σlimits_{m = 0}^{100} {{,^{100}}{Cₘ}{{(x - 3)}^{10

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7.Binomial Theorem

hard

The coefficient of ${x^{53}}$ in the following expansion $\sum\limits_{m = 0}^{100} {{\,^{100}}{C_m}{{(x - 3)}^{100 - m}}} {.2^m}$is

A

$^{100}{C_{47}}$

B

$^{100}{C_{53}}$

C

${ - ^{100}}{C_{53}}$

D

${ - ^{100}}{C_{100}}$

Solution

(c) The given sigma is expansion of ${[(x – 3) + 2]^{100}} = {(x – 1)^{100}} = {(1 – x)^{100}}$.

${x^{53}}$will occur in ${T_{54}}$. ${T_{54}} = {\,^{100}}{C_{53}}{( – x)^{53}}$

Coefficient is $-^{100}C_{53}$.

Standard 11

Mathematics

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