Binomial Theorem

If the coefficients of \(x^4, x^5\) and \(x^6\) in the expansion of \((1+x)^n\) are in the arithmetic progression, then the maximum value of \(n\) is:

In the expansion of \((1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0\), the sum of the coefficients of $x^3$ and \(x^{-13}\) is equal to __________.

Let \(a\) be the sum of all coefficients in the expansion of \(\left(1-2 x+2 x^2\right)^{2023}\left(3-4 x^2+2 x^3\right)^{2024}\) and \(b=\lim _\limits{x \rightarrow 0}\left(\frac{\int_0^x \frac{\log (1+t)}{t^{2024}+1} d t}{x^2}\right)\). I...

Let the coefficient of \(x^r\) in the expansion of \((x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots \ldots \ldots .+(x+2)^{n-1}\) be \(\alpha_r\). If \(\sum_\limits{r=0}^n \alpha_r=\beta^n-\gamma^n, \beta, \gamma \in \mathbb{N}\), t...

\(\text { Number of integral terms in the expansion of }\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}^{824} \text { is equal to _________. }\)

Let \(\alpha=\sum_\limits{k=0}^n\left(\frac{\left({ }^n C_k\right)^2}{k+1}\right)\) and \(\beta=\sum_\limits{k=0}^{n-1}\left(\frac{{ }^n C_k{ }^n C_{k+1}}{k+2}\right)\) If \(5 \alpha=6 \beta\), then \(n\) equals _______.

Suppose \(2-p, p, 2-\alpha, \alpha\) are the coefficients of four consecutive terms in the expansion of \((1+x)^n\). Then the value of \(p^2-\alpha^2+6 \alpha+2 p\) equals

\(\text { If } \frac{{ }^{11} C_1}{2}+\frac{{ }^{11} C_2}{3}+\ldots+\frac{{ }^{11} C_9}{10}=\frac{n}{m} \text { with } \operatorname{gcd}(n, m)=1 \text {, then } n+m \text { is equal to }\) _______.

Remainder when \(64^{32^{32}}\) is divided by 9 is equal to ________.

If A denotes the sum of all the coefficients in the expansion of $\left(1-3 x+10 x^2\right)^{\mathrm{n}}$ and B denotes the sum of all the coefficients in the expansion of $\left(1+x^2\right)^n$, then :

${ }^{n-1} C_r=\left(k^2-8\right){ }^n C_{r+1}$ if and only if :

The coefficient of \(x^{2012}\) in the expansion of \((1-x)^{2008}\left(1+x+x^2\right)^{2007}\) is equal to _________.

If the Coefficient of $x^{30}$ in the expansion of $\left(1+\frac{1}{x}\right)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 ; x \neq 0$ is $\alpha$, then $|\alpha|$ equals ___________.

Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3} x^{\frac{1}{3}}+\frac{1}{2 x^{\frac{2}{3}}}\right)^{18}$. Then $\left(\frac{\mathrm{n}}{\mathrm{m}}\right)^{\frac{1}{3}}...

The largest natural number \(n\) such that \(3^{n}\) divides \(66 !\) is ___________.

Let \([t]\) denote the greatest integer \(\leq t\). If the constant term in the expansion of \(\left(3 x^{2}-\frac{1}{2 x^{5}}\right)^{7}\) is \(\alpha\), then \([\alpha]\) is equal to ___________.

If the coefficients of three consecutive terms in the expansion of \((1+x)^{n}\) are in the ratio \(1: 5: 20\), then the coefficient of the fourth term is

The absolute difference of the coefficients of \(x^{10}\) and \(x^{7}\) in the expansion of \(\left(2 x^{2}+\frac{1}{2 x}\right)^{11}\) is equal to :

\(25^{190}-19^{190}-8^{190}+2^{190}\) is divisible by :

The coefficient of \(x^{18}\) in the expansion of \(\left(x^{4}-\frac{1}{x^{3}}\right)^{15}\) is __________.

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

If \({ }^{2 n} C_{3}:{ }^{n} C_{3}=10: 1\), then the ratio \(\left(n^{2}+3 n\right):\left(n^{2}-3 n+4\right)\) is :

Among the statements :
(S1) : \(2023^{2022}-1999^{2022}\) is divisible by 8
(S2) : \(13(13)^{n}-12 n-13\) is divisible by 144 for infinitely many \(n \in \mathbb{N}\)

If the coefficient of \({x^7}\) in \({\left( {a{x^2} + {1 \over {2bx}}} \right)^{11}}\) and \({x^{ - 7}}\) in \({\left( {ax - {1 \over {3b{x^2}}}} \right)^{11}}\) are equal, then :

Let \(\alpha>0\), be the smallest number such that the expansion of \(\left(x^{\frac{2}{3}}+\frac{2}{x^{3}}\right)^{30}\) has a term \(\beta x^{-\alpha}, \beta \in \mathbb{N}\). Then \(\alpha\) is equal to ___________.

The remainder on dividing \(5^{99}\) by 11 is ____________.

If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{l}}\right)^{9}$ is $-84$ and the coefficient of $x^{-3 l}$ is

$2^{\alpha} \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is e...

The coefficient of $x^{-6}$, in the
expansion of $\left(\frac{4 x}{5}+\frac{5}{2 x^{2}}\right)^{9}$, is

The coefficient of \({x^{301}}\) in \({(1 + x)^{500}} + x{(1 + x)^{499}} + {x^2}{(1 + x)^{498}}\, + \,...\, + \,{x^{500}}\) is :

If the coefficient of \(x^{15}\) in the expansion of \(\left(\mathrm{a} x^{3}+\frac{1}{\mathrm{~b} x^{1 / 3}}\right)^{15}\) is equal to the coefficient of \(x^{-15}\) in the expansion of \(\left(a x^{1 / 3}-\frac{1}{b x^{3}}\right)^{15}\), ...

$50^{\text {th }}$ root of a number $x$ is 12 and $50^{\text {th }}$ root of another number $y$ is 18 . Then the remainder obtained on dividing $(x+y)$ by 25 is ____________.

Let $x=(8 \sqrt{3}+13)^{13}$ and $y=(7 \sqrt{2}+9)^9$. If $[t]$ denotes the greatest integer $\leq t$, then :

If the co-efficient of \(x^9\) in \({\left( {\alpha {x^3} + {1 \over {\beta x}}} \right)^{11}}\) and the co-efficient of \(x^{-9}\) in \({\left( {\alpha x - {1 \over {\beta {x^3}}}} \right)^{11}}\) are equal, then \((\alpha\beta)^2\) is equ...

Let the coefficients of three consecutive terms in the binomial expansion of \((1+2x)^n\) be in the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of those three terms, is __________.

Let K be the sum of the coefficients of the odd powers of \(x\) in the expansion of \((1+x)^{99}\). Let \(a\) be the middle term in the expansion of \({\left( {2 + {1 \over {\sqrt 2 }}} \right)^{200}}\). If $${{{}^{200}{C_{99}}K} \over a} =...

The constant term in the expansion of \({\left( {2x + {1 \over {{x^7}}} + 3{x^2}} \right)^5}\) is ___________.

If \(a_r\) is the coefficient of \(x^{10-r}\) in the Binomial expansion of \((1 + x)^{10}\), then \(\sum\limits_{r = 1}^{10} {{r^3}{{\left( {{{{a_r}} \over {{a_{r - 1}}}}} \right)}^2}}\) is equal to

The remainder when (2023)\(^{2023}\) is divided by 35 is __________.

Suppose \(\sum\limits_{r = 0}^{2023} {{r^2}{}~^{2023}{C_r} = 2023 \times \alpha \times {2^{2022}}}\). Then the value of \(\alpha\) is ___________

The value of \(\sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_r}}\) is

Let the sum of the coefficients of the first three terms in the expansion of \({\left( {x - {3 \over {{x^2}}}} \right)^n},x \ne 0.~n \in \mathbb{N}\), be 376. Then the coefficient of \(x^4\) is __________.

If \({({}^{30}{C_1})^2} + 2{({}^{30}{C_2})^2} + 3{({}^{30}{C_3})^2}\, + \,...\, + \,30{({}^{30}{C_{30}})^2} = {{\alpha 60!} \over {{{(30!)}^2}}}\) then \(\alpha\) is equal to :

The remainder, when \(19^{200}+23^{200}\) is divided by 49 , is ___________.

If the term without \(x\) in the expansion of \(\left(x^{\frac{2}{3}}+\frac{\alpha}{x^{3}}\right)^{22}\) is 7315 , then \(|\alpha|\) is equal to ___________.

Let the sixth term in the binomial expansion of \({\left( {\sqrt {{2^{{{\log }_2}\left( {10 - {3^x}} \right)}}} + \root 5 \of {{2^{(x - 2){{\log }_2}3}}} } \right)^m}\) in the increasing powers of \(2^{(x-2) \log _{2} 3}\), be 21 . If the ...

Let $\left(a+b x+c x^{2}\right)^{10}=\sum\limits_{i=0}^{20} p_{i} x^{i}, a, b, c \in \mathbb{N}$. If $p_{1}=20$ and $p_{2}=210$, then

$2(a+b+c)$ is equal to :

Let \(\alpha\) be the constant term in the binomial expansion of \(\left(\sqrt{x}-\frac{6}{x^{\frac{3}{2}}}\right)^{n}, n \leq 15\). If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $$x^{-...

Fractional part of the number \(\frac{4^{2022}}{15}\) is equal to

The remainder, when \(7^{103}\) is divided by 17, is __________

The coefficient of \(x^{5}\) in the expansion of \(\left(2 x^{3}-\frac{1}{3 x^{2}}\right)^{5}\) is :