(A) The $(r+1)^{\text{th}}$ term,$T_{r+1}$,in the binomial expansion of $(a+b)^n$ is given by $T_{r+1} = {^nC_r} a^{n-r} b^r$.For the expansion of $(x

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(A) The $(r+1)^{\text{th}}$ term,$T_{r+1}$,in the binomial expansion of $(a+b)^n$ is given by $T_{r+1} = {^nC_r} a^{n-r} b^r$.For the expansion of $(x-2y)^{12}$,we have $n=12$,$a=x$,and $b=-2y$.The $4^{\text{th}}$ term corresponds to $r+1=4$,so $r=3$.$T_4 = T_{3+1} = {^{12}C_3} (x)^{12-3} (-2y)^3$$T_4 = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} \times x^9 \times (-8)y^3$$T_4 = 220 \times x^9 \times (-8)y^3 = -1760x^9y^3$.

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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Find the $4^{\text{th}}$ term in the expansion of $(x-2y)^{12}$. (in $x^9y^3$)

A
$-1760$
B
$1760$
C
$-17600$
D
$17600$

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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

Let $n$ be a positive integer. If the coefficients of $2^{\text{nd}}$,$3^{\text{rd}}$,and $4^{\text{th}}$ terms in the expansion of $(1+x)^n$ are in $A$.$P$.,then the value of $n$ is:

EasyAP EAMCET 2020

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The least value of $n$ for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7}+\sqrt[12]{11})^{n}$ is $183$ is:

DifficultJEE MAIN 2025

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If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of $\left(\frac{1}{x^3} - x^4\right)^n, x \neq 0$, is zero, then the value of $n$ is . . . . . . .

DifficultJEE MAIN 2026

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If $n$ is an even positive integer,then the condition that the greatest term in the expansion of $(1 + x)^n$ may have the greatest coefficient also,is

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The $16^{th}$ term in the expansion of $(\sqrt{x} - \sqrt{y})^{17}$ is

Easy

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Find the $4^{\text{th}}$ term in the expansion of $(x-2y)^{12}$. (in $x^9y^3$)

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Let $n$ be a positive integer. If the coefficients of $2^{\text{nd}}$,$3^{\text{rd}}$,and $4^{\text{th}}$ terms in the expansion of $(1+x)^n$ are in $A$.$P$.,then the value of $n$ is:

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