(B) The general term in the expansion of $(1 + x)^n$ is given by $T_{k+1} = {}^nC_k x^k$.For the expansion of $(1 + x)^{21}$,the coefficient of $x^r$

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(B) The general term in the expansion of $(1 + x)^n$ is given by $T_{k+1} = {}^nC_k x^k$.For the expansion of $(1 + x)^{21}$,the coefficient of $x^r$ is ${}^{21}C_r$ and the coefficient of $x^{r+1}$ is ${}^{21}C_{r+1}$.Given that the coefficients are equal,we have ${}^{21}C_r = {}^{21}C_{r+1}$.Using the property ${}^nC_a = {}^nC_b \Rightarrow a = b$ or $a + b = n$,we get $r + (r + 1) = 21$.$2r + 1 = 21$ $\Rightarrow 2r = 20$ $\Rightarrow r = 10$.

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

Easy

If in the expansion of $(1 + x)^{21}$,the coefficients of $x^r$ and $x^{r + 1}$ are equal,then $r$ is equal to

A
$9$
B
$10$
C
$11$
D
$12$

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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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MediumAP EAMCET 2023

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EasyTS EAMCET 2021

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The term independent of $x$ in ${\left[ {\sqrt{\frac{x}{3}} + \frac{{\sqrt{3}}}{{{x^2}}}} \right]^{10}}$ is

Easy

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The index of the power of $x$ occurring in the $5^{\text{th}}$ term from the end in the expansion of $\left(\frac{x^3}{2} - \frac{2}{x^2}\right)^{12}$ is

EasyAP EAMCET 2020

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If in the expansion of $(1 + x)^{21}$,the coefficients of $x^r$ and $x^{r + 1}$ are equal,then $r$ is equal to

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