(C) The general term ${T_k}$ in the expansion of ${(a + b)^n}$ is given by ${T_{k+1} = {^nC_k} a^{n-k} b^k}$.Here,$n = 15$,$a = x^4$,and $b = x^{-3}$.

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(C) The general term ${T_k}$ in the expansion of ${(a + b)^n}$ is given by ${T_{k+1} = {^nC_k} a^{n-k} b^k}$.Here,$n = 15$,$a = x^4$,and $b = x^{-3}$.The ${r^{th}}$ term is ${T_r = T_{(r-1)+1} = {^{15}C_{r-1}} (x^4)^{15-(r-1)} (x^{-3})^{r-1}}$.${T_r = {^{15}C_{r-1}} (x^4)^{16-r} (x^{-3})^{r-1} = {^{15}C_{r-1}} x^{64-4r} x^{-3r+3} = {^{15}C_{r-1}} x^{67-7r}}$.Given that the term contains ${x^4}$,we set the exponent of $x$ equal to $4$:$67 - 7r = 4$.$7r = 63$.$r = 9$.

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

Medium

If ${x^4}$ occurs in the ${r^{th}}$ term in the expansion of ${\left( {{x^4} + \frac{1}{{{x^3}}}} \right)^{15}}$,then $r = $

A
$7$
B
$8$
C
$9$
D
$10$

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Class 11

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Chapter

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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The middle term in the expansion of ${\left( {x + \frac{1}{{2x}}} \right)^{2n}}$ is:

Easy

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If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $(\sqrt[4]{2} + \frac{1}{\sqrt[4]{3}})^n$ is $\sqrt{6} : 1$,then the third term from the beginning is:

DifficultJEE MAIN 2023

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If coefficients of $2^{nd}$,$3^{rd}$ and $4^{th}$ terms in the binomial expansion of $(1 + x)^n$ are in $A.P.$,then $n^2 - 9n$ is equal to

Medium

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If the coefficients of the $p^{th}$,$(p + 1)^{th}$,and $(p + 2)^{th}$ terms in the expansion of $(1 + x)^n$ are in $A.P.$,then

MediumAIEEE 2005

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If ${x^4}$ occurs in the ${r^{th}}$ term in the expansion of ${\left( {{x^4} + \frac{1}{{{x^3}}}} \right)^{15}}$,then $r = $

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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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