Binomial Theorem Questions - JEE

Find the coefficient of $x^3$ in the expansion of $(1+2x)^{-3}$ when \(|x|

The value of r for which $^{20} \mathrm{C}_{r} \ ^{20} \mathrm{C}_{0}+\ ^{20} \mathrm{C}_{r-1}\ ^{20} \mathrm{C}_{1}+\ ^{20} \mathrm{C}_{r-2}\ ^{20} \mathrm{C}_{2}+\ldots .+\ ^{20} \mathrm{C}_{0}\ ^{20} \mathrm{C}_{r}$ is maximum, is

If $n$ is the degree of the polynomial, \[\left[\dfrac{2}{\sqrt{5x^3+1}-\sqrt{5x^3-1}}\right]^8+\left[\dfrac{2}{\sqrt{5x^3+1}+\sqrt{5x^3-1}}\right]^8\] and $m$ is the coefficient of $x^n$ in it, then the ordered pair $(n,m)$ is equal to

If the term independent of $x$ in the expansion of $\left(\sqrt{ax^2}+\dfrac{1}{2x^3}\right)^{10}$ is 105, then $a^2$ is equal to:

Find the fourth term in the expansion of $(1-2x)^{-3/4}$ if \(|x|

If $\dfrac{{}^{11}C_1}{2} + \dfrac{{}^{11}C_2}{3} + \ldots + \dfrac{{}^{11}C_9}{10} = \dfrac{n}{m}$ with $\gcd(n,m)=1$, then $n+m$ is equal to

The value of $\sum_\limits{r=0}^{n}$ nCr (sin rx) is equal to

If $x = 1 + 3a + 6a^2 + 10a^3 + \ldots$ to $\infty$, $|a| \(y = 1 + 4b + 10b^2 + 20b^3 + \ldots$ to $\infty$, $|b| and \(S = 1 + 3(ab) + 5(ab)^2 + \ldots$ to $\infty$,then find $S$ in terms of $x$ and $y$.

The value of $\dfrac{{}^{100}C_{50}}{51}+\dfrac{{}^{100}C_{51}}{52}+\cdots+\dfrac{{}^{100}C_{100}}{101}$ is:

Let $S=\dfrac{1}{2!\,23!}+\dfrac{1}{3!\,22!}+\dfrac{1}{4!\,21!}+\cdots$ up to 13 terms. If $13S=\dfrac{2^k}{n!}$, $k\in\mathbf{N}$, then $n+k$ is equal to

If $\displaystyle\sum_{r=0}^{2n} a_r(x-2)^r = \sum_{r=0}^{2n} b_r(x-3)^r$ and $a_k = 1$ for all $k \ge n$, then $b_n$ is equal to

The number of integer terms in the expansion of $(\sqrt{3} + \sqrt[8]{5})^{256}$ is:

If $a = \displaystyle\sum_{r=0}^{n} \dfrac{1}{{}^nC_r}$, then the value of \(\displaystyle\sum_{0 \le i

The number of integral terms in the expansion of $(5^{1/2} + 7^{1/8})^{1024}$ is

If terms independent of $x$ in the expansion of $\left(3x - \dfrac{1}{x}\right)^{20}$ and $\left(x + \dfrac{\sqrt[9]{3^{10}}}{x}\right)^{18}$ are $A$ and $B$ respectively, then $\left(\dfrac{9}{38}A + B\right)$ equals:

Find the coefficient of $x^9 y^{-3}$ in the expansion of $\left(\dfrac{2x^2}{y} + \dfrac{y}{3x}\right)^{12}$.

The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+\cdots+x^{54}(1+x)^{46}$ is $\binom{99}{p}-\binom{46}{q}$. Then a possible value of $p+q$ is:

Let S be the sum of the digits of the coefficient of x6 in the expansion of (1 + 2x - 3x2)4. Then which of the following statements is not correct?

Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1+x)^n$, $n\in\mathbf{N}$, $0\leq r\leq n$. If $P_n=C_0-C_1+\dfrac{2^2}{3}C_2-\dfrac{2^3}{4}C_3+\cdots+\dfrac{(-2)^n}{n+1}C_n$, then the value of $\displaystyle\sum_{n=1}^{25}\dfrac{1}{P_{2n}}$ equals.

Given $\left(\dfrac{1}{60} - \dfrac{x^8}{81}\right)\left(2x^2 - \dfrac{3}{x^2}\right)^6$, find the coefficient of $x^0$.

Let $a = \sqrt{5x^3+1}$ and $b = \sqrt{5x^3-1}$. If $\left[\dfrac{2(\sqrt{5x^3+1})+\sqrt{5x^3-1}}{2}\right]^8 + \left[\dfrac{2(\sqrt{5x^3+1})-\sqrt{5x^3-1}}{2}\right]^8 = \text{polynomial of degree } n$ with coefficient of $x^n$ equal to $m$, then $(n, m)$ equals:

If the number of terms in the expansion of $\left(1 - \dfrac{2}{x} + \dfrac{4}{x^2}\right)^n$, $x \neq 0$, is 28, then the sum of the coefficients of all terms in this expansion, is

The value of $\dfrac{1}{1!50!}+\dfrac{1}{3!48!}+\dfrac{1}{5!46!}+\cdots+\dfrac{1}{49!2!}+\dfrac{1}{51!1!}$ is:

The remainder on dividing $5^{99}$ by 11 is ___.

The sum of the coefficients of $x^{2/3}$ and $x^{-2/5}$ in the binomial expansion of $\left(x^{2/3}+\dfrac{1}{2}x^{-2/5}\right)^9$ is:

If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1+x)^{10}$, then $\displaystyle\sum_{r=1}^{10}r^3\left(\dfrac{a_r}{a_{r-1}}\right)^2$ is equal to:

The value of $\displaystyle\sum_{r=1}^{n+1}\left(\sum_{k=1}^{n} {}^k C_{r-1}\right)$ (where $r, k, n \in \mathbb{N}$) is equal to

The last digit in 7300 is

If $(1 + x)^5 = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5$, then the value of $(a_0 - a_2 + a_4)^2 + (a_1 - a_3 + a_5)^2$ is equal to

Find the term independent of x in the expansion of $\left(3x - \dfrac{1}{x}\right)^{20}$. If this term is A, then find the term independent of x in the expansion of $\left(x + \dfrac{\sqrt[9]{3^{10}}}{x}\right)^{18}$.

The remainder when (2021)2022 + (2022)2021 is divided by 7 is

The smallest natural number $n$, such that the coefficient of $x$ in the expansion of $\left(x^2 + \dfrac{1}{x^3}\right)^n$ is ${}^nC_{23}$, is ___________.

If the coefficient of the middle term in the expansion of $(1+x)^{2n+2}$ is $\alpha$ and the coefficients of middle terms in the expansion of $(1+x)^{2n+1}$ are $\beta$ and $\gamma$, then relate $\alpha$, $\beta$, and $\gamma$.

The remainder left out when $8^{2n} - (62)^{2n+1}$ is divided by 9 is

The coefficient of $x^n$ in the expansion of $(1 + x + x^2 + x^3 + \cdots)^2$ is:

If n is even, the value of ∑r=0n/2 - 1 a2r is

If the coefficients of $x^{-2}$ and $x^{-4}$ in the expansion of $\left(x^{1/3} + \dfrac{1}{2x^{1/3}}\right)^{18}$, $(x > 0)$, are $m$ and $n$, respectively, then $\dfrac{m}{n}$ is equal to

Let Tr be the rth term of a sequence, for r = 1, 2, 3, ... If 3Tr+1 = Tr and T7 = $\frac{1}{243}$, then the value of $\sum_{r=1}^{\infty}\left(T_{r} \cdot T_{r+1}\right)$ is:

If the middle term of $\left(\dfrac{1}{x} + x\sin x\right)^{10}$ is equal to $7\dfrac{7}{8}$, then find the value of $x$.

Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2 + (1+x)^3 + \ldots + (1+x)^{49} + (1+mx)^{50}$ is $(3n+1)\,{}^{51}C_3$ for some positive integer $n$. Then the value of $n$ is _______.

For $x\in\mathbb{R}$, $x\neq -1$, if $(1+x)^{2016}+x(1+x)^{2015}+x^2(1+x)^{2014}+\cdots+x^{2016}=\displaystyle\sum_{i=0}^{2016}a_ix^i$, then $a_{17}$ is equal to

The sum of rational terms in $(\sqrt{2} + \sqrt[3]{3} + \sqrt[5]{5})^{10}$ is equal to

If $a_n = \displaystyle\sum_{r=0}^{n} \dfrac{1}{^nC_r} = b_n = \displaystyle\sum_{r=0}^{n} \dfrac{1}{^nC_r}$, then the number of ordered pairs $(p, q)$ such that $c_p + c_q = 1$, where $c_p = \dfrac{a_p}{b_p}$, is:

The coefficient of $x^2$ in the expansion of $\frac{(1+x)^{3/2} - \left(1 + \frac{1}{2}x\right)^3}{(1-x)^{1/2}}$ is:

The expansion $a^m\left\{\left(1 + \dfrac{b}{a}\right)^m\right\}$ is valid only when:

If the middle term of $\left(\frac{1}{x} + x \sin x\right)^{10}$ is equal to $7\frac{7}{8}$, then value of $x$ is

If in the expansion of $(1+x)^n$, $a, b, c$ are three consecutive coefficients, then $n =$

If $x > 0$, the first negative term in the expansion of $(1+x)^{27/5}$ is:

The sum of the last eight coefficients in the expansion of $(1 + x)^{16}$ is equal to

The value of $\dfrac{{}^nC_0}{n} + \dfrac{{}^nC_1}{n+1} + \dfrac{{}^nC_2}{n+2} + \cdots + \dfrac{{}^nC_n}{2n}$ is equal to

Binomial Theorem Questions (605)