(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

For some $n \neq 10$,let the coefficients of the $5^{\text{th}}$,$6^{\text{th}}$,and $7^{\text{th}}$ terms in the binomial expansion of $(1+x)^{n+4}$ be in $A.P.$ Then the largest coefficient in the expansion of $(1+x)^{n+4}$ is:

DifficultJEE MAIN 2025

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MediumJEE MAIN 2021

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The total number of terms in the expansion of $(x + a)^{100} + (x - a)^{100}$ after simplification will be

Easy

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If $n$ is a positive integer and $f(n)$ is the coefficient of $x^n$ in the expansion of $(1+x)(1-x)^n$,then $f(2023) = $

MediumTS EAMCET 2023

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The smallest natural number $n$ such that the coefficient of $x$ in the expansion of $(x^2 + \frac{1}{x^3})^n$ is $^nC_{23}$ is

DifficultJEE MAIN 2019

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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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For some $n \neq 10$,let the coefficients of the $5^{\text{th}}$,$6^{\text{th}}$,and $7^{\text{th}}$ terms in the binomial expansion of $(1+x)^{n+4}$ be in $A.P.$ Then the largest coefficient in the expansion of $(1+x)^{n+4}$ is:

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