The coefficient of x256 in the expansion of
(1 $-$ x)101 (x2 + x + 1)100 is :
Solution
${(1 - x)^{101}}{({x^2} + x + 1)^{100}}$<br/><br/>Coefficient of ${x^{256}} = {[(1 - x)(1 + x + {x^2})]^{100}}(1 - x) = {(1 - {x^3})^{100}}(1 - x)$<br/><br/>$$ \Rightarrow ({}^{100}{C_0} - {}^{100}{C_1}{x^3} + {}^{100}{C_2}{x^6} - {}^{100}{C_3}{x^9}...)(1 - x)$$<br/><br/>$\sum {{{( - 1)}^r}{}^{100}{C_r}{x^{3r}}(1 - x)}$<br/><br/>$\Rightarrow 3r = 256$ or $255 \Rightarrow r = {{256} \over 3}$ (Reject)<br/><br/>r = 85<br/><br/>Coefficient = ${}^{100}{C_{85}} = {}^{100}{C_{15}}$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Applications of Binomial Theorem
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