(A) Given expression: $(1+x)(1-x)^{10}(1+x+x^2)^9$We know that $(1-x)(1+x+x^2) = (1-x^3)$.Rewrite the expression as: $(1-x)(1-x^2)(1-x^3)^9$$= (1-x-x^

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(A) Given expression: $(1+x)(1-x)^{10}(1+x+x^2)^9$We know that $(1-x)(1+x+x^2) = (1-x^3)$.Rewrite the expression as: $(1-x)(1-x^2)(1-x^3)^9$$= (1-x-x^2+x^3)(1-x^3)^9$$= (1-x-x^2+x^3) \sum_{k=0}^{9} \binom{9}{k} (1)^ {9-k} (-x^3)^k$$= (1-x-x^2+x^3) \sum_{k=0}^{9} \binom{9}{k} (-1)^k x^{3k}$We need the coefficient of $x^{18}$. This occurs when $3k = 18$,i.e.,$k=6$.The term corresponding to $k=6$ is $\binom{9}{6} (-1)^6 x^{18} = 84 x^{18}$.Thus,the coefficient is $84$.

Class 11 Mathematics Binomial Theorem General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

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The coefficient of $x^{18}$ in the product $(1+ x)(1- x)^{10} (1+ x + x^2 )^9$ is

JEE MAIN 2019 All JEE MAIN

A
$84$
B
$126$
C
$-126$
D
$-84$

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Binomial Theorem

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General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

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