(C) The expression is $(1 + x + x^2 + \dots + x^{100})^3 = (\frac{1-x^{101}}{1-x})^3 = (1-x^{101})^3(1-x)^{-3}$.Expanding this,we get $(1 - 3x^{101} +

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(C) The expression is $(1 + x + x^2 + \dots + x^{100})^3 = (\frac{1-x^{101}}{1-x})^3 = (1-x^{101})^3(1-x)^{-3}$.Expanding this,we get $(1 - 3x^{101} + 3x^{202} - x^{303}) \sum_{n=0}^{\infty} \binom{n+3-1}{3-1} x^n$.We are looking for the coefficient of $x^{100}$.From the expansion,only the term $1 \times \binom{100+2}{2} x^{100}$ contributes to $x^{100}$.Thus,the coefficient is $\binom{102}{2} = ^{102}C_2$.

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

What is the coefficient of $x^{100}$ in $(1 + x + x^2 + x^3 + \dots + x^{100})^3$?

A
$^{100}C_3$
B
$^{102}C_3$
C
$^{102}C_2$
D
$^{105}C_2$

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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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What is the coefficient of $x^{100}$ in $(1 + x + x^2 + x^3 + \dots + x^{100})^3$?

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