Binomial Theorem — JEE Main PYQs

The remainder when $(2023{)}^{2023}$ is divided by 35 is ______.

If ${\left({}^{30}{C}_1\right)}^2+2{\left({}^{30}{C}_2\right)}^2+3{\left({}^{30}{C}_3\right)}^2+\dots +30{\left({}^{30}{C}_{30}\right)}^2=\frac{\alpha \cdot 60!}{(30!{)}^2}$, then $\alpha$ is equal to

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

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

If the constant term in the binomial expansion of ${\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{\ell }}\right)}^9$ is -84 and the Coefficient of $x^{-3\ell }$ is $2^{\alpha }\beta$, where $\beta <0$ is an odd number, Then $|\alpha \ell -\beta |$ is equal to

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

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

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

The value of
$\frac{1}{1!50!}+\frac{1}{3!48!}+\frac{1}{5!46!}+\dots .+\frac{1}{49!2!}+\frac{1}{51!1!}$ is

The remainder when $19^{200}+23^{200}$ is divided by 49 , is ______.

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

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

If ${\sum }_{\mathrm{k}=1}^{10}{\mathrm{K}}^2{\left({}^{10}{\mathrm{C}}_{\mathrm{K}}\right)}^2=22000\mathrm{L}$, then L is equal to ____.

The remainder when $7^{2022}+3^{2022}$ is divided by 5 is:

Let the coefficients of the middle terms in the expansion of ${\left(\frac{1}{\sqrt{6}}+\beta x\right)}^4,(1-3\beta x{)}^2$ and ${\left(1-\frac{\beta }{2}x\right)}^6,\beta >0$, respectively form the first three terms of an A.P. If d is the common difference of this A.P., the …

If $1+\left(2+{}^{49}{\mathrm{C}}_1+{}^{49}{\mathrm{C}}_2+\dots ..+{}^{49}{\mathrm{C}}_{49}\right)\left({}^{50}{\mathrm{C}}_2+{}^{50}{\mathrm{C}}_4+\dots ..+{}^{50}{\mathrm{C}}_{50}\right)$ is equal to $2^{\mathrm{n}}.\mathrm{m}$, where m is odd, then n +m is eq …

The remainder when $(2021{)}^{2022}+(2022{)}^{2021}$ is divided by 7 is

If the maximum value of the term independent of t in the expansion of ${\left(t^2{x}^{\frac{1}{5}}+\frac{(1-x{)}^{\frac{1}{10}}}{t}\right)}^{15}$, $x\ge 0$, is K, then 8K is equal to ____.

${\sum }_{\begin{array}{c}i,j=0\\ i\ne j\end{array}}^n{}^n{C}_i{}^n{C}_j$ is equal to

Let for the $9^{\text{th}}$ term in the binomial expansion of $(3+6x{)}^n$, in the increasing powers of $6x$, to be the greatest for $x=\frac{3}{2}$, the least value of n is $n_0$. If k is the ratio of the coefficient of $x^6$ to the coefficient of $x^3$, then $k+n_0$ is …

The remainder when $(11{)}^{1011}+(1011{)}^{11}$ is divided by 9 is

If the coefficients of x and $x^2$ in the expansion of $(1+x{)}^p(1-x{)}^q,p,q\le 15$, are -3 and -5 respectively, then the coefficient of $x^3$ is equal to ______.

The term independent of $x$ in the expression of $\left(1-x^2+3x^3\right){\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11},x\ne 0$ is

If the sum of the coefficients of all the positive powers of x, in the binomial expansion of ${\left(x^n+\frac{2}{x^5}\right)}^7$ is 939, then the sum of all the possible integral values of n is :

If

Where …

The number of positive integers k such that the constant term in the binomial expansion of ${\left(2x^3+\frac{3}{x^k}\right)}^{12}$, $x\ne 0$ is $2^8\cdot \ell$, where $\ell$ is an odd integer, is ____.

Let $n\ge 5$ be an integer. If $9^n-8n-1=64\alpha$ and $6^{\mathrm{n}}-5\mathrm{n}-1=25\beta$, then $\alpha -\beta$ is equal to:

Let the coefficients of ${\mathrm{x}}^{-1}$ and ${\mathrm{x}}^{-3}$ in the expansion of ${\left(2x^{\frac{1}{5}}-\frac{1}{x^{\frac{1}{5}}}\right)}^{15},\mathrm{x}>0$, be $m$ and $n$ respectively. If r is a positive integer such that …

If $\left({}^{40}{\mathrm{C}}_0\right)+\left({}^{41}{\mathrm{C}}_1\right)+\left({}^{42}{\mathrm{C}}_2\right)+\dots +\left({}^{60}{\mathrm{C}}_{20}\right)=\frac{\mathrm{m}}{\mathrm{n}}{}^{60}{\mathrm{C}}_{20}$, $m$ and $n$ are coprime, then $m+n$ is equal to ____.

If the constant term in the expansion of ${\left(3x^3-2x^2+\frac{5}{x^5}\right)}^{10}$ is $2^{\mathrm{k}}.l$, where $l$ is an odd integer, then the value of k is equal to :

The remainder when $(2021{)}^{2023}$ is divided by 7 is :

If the coefficient of $x^{10}$ in the binomial expansion of ${\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}}+\frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)}^{60}$ is $5^{k}l$, where $l,k\in \mathbb{N}$ and $l$ is coprime to 5, then k is equal to ______.

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of ${\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)}^n$, in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6}:1$. If the sixth term from the beginning …

The remainder when $3^{2022}$ is divided by 5 is

The remainder on dividing $1+3+3^2+3^3+\dots +3^{2021}$ by 50 is ____.

The coefficient of $x^{101}$ in the expression $(5+x{)}^{500}+x(5+x{)}^{499}+x^2(5+x{)}^{498}+\dots ..x^{500}$, $\mathrm{x}>0$, is

If the sum of the coefficients of all the positive even powers of x in the binomial expansion of ${\left(2x^3+\frac{3}{x}\right)}^{10}$ is $5^{10}-\beta \cdot 3^9$, then $\beta$ is equal to

If ${}^{20}{\mathrm{C}}_{\mathrm{r}}$ is the co-efficient of ${\mathrm{x}}^{\mathrm{r}}$ in the expansion of $(1+\mathrm{x}{)}^{20}$, then the value of ${\sum }_{\mathrm{r}=0}^{20}{\mathrm{r}}^2{}^{20}{\mathrm{C}}_{\mathrm{r}}$ is equal to

The value of $-{}^{15}{\mathrm{C}}_1+2.{}^{15}{\mathrm{C}}_2-3.{}^{15}{\mathrm{C}}_3+\dots ..-15.{}^{15}{\mathrm{C}}_{15}+{}^{14}{\mathrm{C}}_1+{}^{14}{\mathrm{C}}_3+{}^{14}{\mathrm{C}}_5+\dots .+{}^{14}{\mathrm{C}}_{11}$ is :

Let $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ denote ${}^n{C}_k$ and $\left[\begin{array}{c}n\\ k\end{array}\right]=\{\begin{array}{l}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\text{, if }0\le k\le n\\ 0\text{, otherwise }\end{array}$ If …

The coefficient of $x^{256}$ in the expansion of $(1-x{)}^{101}{\left(x^2+x+1\right)}^{100}$ is :

The number of rational terms in the binomial expansion of ${\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)}^{120}$ is ______.

If $\mathrm{n}\ge 2$ is a positive integer, then the sum of the series ${}^{\mathrm{n}+1}{\mathrm{C}}_2+2\left({}^2{\mathrm{C}}_2+{}^3{\mathrm{C}}_2+{}^4{\mathrm{C}}_2+\dots +{}^{\mathrm{n}}{\mathrm{C}}_2\right)$ is:

For integers $n$ and $r$, let $\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\{\begin{array}{ll}n{C}_r,& \text{ if }n\ge r\ge 0\\ 0,& \text{ otherwise }\end{array}$
The maximum value of k for which the sum …

For the natural numbers $m$, $n$, if $(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+\dots .+a_{m+n}{y}^{m+n}$ and $a_1=a_2=10$, then the value of $(m+n)$ is equal to :

For $k\in N$, let $\frac{1}{\alpha (\alpha +1)(\alpha +2)\dots (\alpha +20)}={\sum }_{k=0}^{20}\frac{A_k}{\alpha +k}$, where $\alpha >0$. Then the value of $100{\left(\frac{A_{14}+A_{15}}{A_{13}}\right)}^2$ is equal to

$3\times 7^{22}+2\times 10^{22}-44$ when divided by 18 leaves the remainder ______.

If the constant term, in binomial expansion of ${\left(2x^r+\frac{1}{x^2}\right)}^{10}$ is 180, then $r$ is equal to ____

The maximum value of term independent of ' t ' in the expansion of ${\left(\mathrm{t}{\mathrm{x}}^{\frac{1}{5}}+\frac{(1-\mathrm{x}{)}^{\frac{1}{10}}}{\mathrm{t}}\right)}^{10}$ where $\mathrm{x}\in (0,1)$ is :