Binomial Theorem — JEE Main PYQs

If for $3 \leq r \leq 30$, $\binom{30}{30-r} + 3\binom{30}{31-r} + 3\binom{30}{32-r} + \binom{30}{33-r} = {}^{m}C_r$, then $m$ equals :

Let the smallest value of $k \in \mathbb{N}$, for which the coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + (1+x)^5 + \ldots + (1+x)^{99} + (1+kx)^{100}$, $x \neq 0$, is $\left(43n + \frac{101}{4}\right)\left({}^{100}C_3\right)$ for some $n \in \mathbb{N}$, be $p$. Then the value of $p + n$ is:

In the expansion of $\left(9x - \dfrac{1}{3\sqrt{x}}\right)^{18}$, $x > 0$, if the term independent of $x$ is $(221)k$, then $k$ is equal to:

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 ________.

The coefficient of $x^2$ in the expansion of $\left(2x^2 + \dfrac{1}{x}\right)^{10}$, $x \neq 0$, is :

If the coefficients of the middle terms in the binomial expansions of $(1+\alpha x)^{26}$ and $(1-\alpha x)^{28}$, $\alpha \neq 0$, are equal, then the value of $\alpha$ is:

If $(1 - x^3)^{10} = \sum_{r=0}^{10} a_r x^r (1 - x)^{30 - 2r}$, then $\frac{9a_9}{a_{10}}$ is equal to ________.

If $26\left(\frac{2^3}{3}\left({}^{12}C_2\right) + \frac{2^5}{5}\left({}^{12}C_4\right) + \frac{2^7}{7}\left({}^{12}C_6\right) + \cdots + \frac{2^{13}}{13}\left({}^{12}C_{12}\right)\right) = 3^{13} - \alpha$, then $\alpha$ is equal to :

The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{\left(x^{2/3} + 1 - x^{1/3}\right)} - \frac{(x-1)}{\left(x - x^{1/2}\right)}\right)^{10}$, $x > 1$, is :

If $\displaystyle\sum_{r=0}^{10}\left(\frac{10^{r+1}-1}{10^r}\right) \cdot {}^{11}C_{r+1}=\frac{\alpha^{11}-11^{11}}{10^{10}}$, then $\alpha$ is equal to :

If $\sum_{r=1}^{9}\left(\frac{r+3}{2^r}\right) \cdot {}^{9}C_r = \alpha\left(\frac{3}{2}\right)^{9} - \beta$, $\alpha, \beta \in \mathbb{N}$, then $(\alpha + \beta)^2$ is equal to

The sum of all rational terms in the expansion of $\left(2 + \sqrt{3}\right)^{8}$ is

Let $(1 + x + x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \ldots + a_{20} x^{20}$. If $(a_1 + a_3 + a_5 + \ldots + a_{19}) - 11a_2 = 121k$, then $k$ is equal to ____________.

In the expansion of $\left(\sqrt[3]{2} + \frac{1}{\sqrt[3]{3}}\right)^n$, $n \in \mathbb{N}$, if the ratio of $15^{\text{th}}$ term from the beginning to the $15^{\text{th}}$ term from the end is $\frac{1}{6}$, then the value of $^{n}\mathrm{C}_3$ is

For an integer $n \ge 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x + y)^{2n-3}$ is 16, then the distance of the point $\mathrm{P}(2n - 1, n^2 - 4n)$ from the line $x + y = 8$ is

If $1^2\cdot\left({}^{15}\mathrm{C}_1\right)+2^2\cdot\left({}^{15}\mathrm{C}_2\right)+3^2\cdot\left({}^{15}\mathrm{C}_3\right)+\ldots+15^2\cdot\left({}^{15}\mathrm{C}_{15}\right)=2^{\mathrm{m}}\cdot 3^{\mathrm{n}}\cdot 5^{\mathrm{k}}$, where $\mathrm{m},\mathrm{n},\mathrm{k}\in\mathbf{N}$, then $\mathrm{m}+\mathrm{n}+\mathrm{k}$ is equal to :

The remainder when $\left((64)^{(64)}\right)^{(64)}$ is divided by 7 is equal to

The sum of the series $2 \times 1 \times {}^{20}C_4 - 3 \times 2 \times {}^{20}C_5 + 4 \times 3 \times {}^{20}C_6 - 5 \times 4 \times {}^{20}C_7 + \cdots + 18 \times 17 \times {}^{20}C_{20}$, is equal to _________.

The number of integral terms in the expansion of $\left(5^{\frac{1}{2}} + 7^{\frac{1}{8}}\right)^{1016}$ is

The product of the last two digits of $(1919)^{1919}$ is __________

If $\sum_{r=0}^{5} \frac{{}^{11}C_{2r+1}}{2r+2}=\frac{m}{n}$, $\gcd(m, n)=1$, then $m-n$ is equal to ________.

Let $\alpha$, $\beta$, $\gamma$ and $\delta$ be the coefficients of $x^7$, $x^5$, $x^3$ and $x$ respectively in the expansion of $\left(x + \sqrt{x^3 - 1}\right)^5 + \left(x - \sqrt{x^3 - 1}\right)^5$, $x > 1$. If $u$ and $v$ satisfy the equations

$\alpha u + \beta v = 18$,

$\gamma u + \delta v = 20$,

then $u + v$ equals :

If $\displaystyle\sum_{r=1}^{30} \dfrac{r^2 \left({}^{30}C_r\right)^2}{{}^{30}C_{r-1}} = \alpha \times 2^{29}$, then $\alpha$ is equal to ________.

The sum of all rational terms in the expansion of $(1 + 2^{\frac{1}{3}} + 3^{\frac{1}{2}})^6$ is equal to ______

If in the expansion of $(1+x)^{\mathrm{p}} (1-x)^{\mathrm{q}}$, the coefficients of $x$ and $x^2$ are 1 and $-2$, respectively, then $\mathrm{p}^2 + \mathrm{q}^2$ is equal to :

For some $n \neq 10$, let the coefficients of the 5th, 6th and 7th 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:

Suppose A and B are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1+x)^{2n-1}$. If $2\mathrm{A} = 5\mathrm{B}$, then $n$ is equal to :

If $\alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} \, ^{12}C_{2r-1}$, then the distance of the point $(12, \sqrt{3})$ from the line $\alpha x - \sqrt{3} y + 1 = 0$ is ____.

Let the coefficients of three consecutive terms $\mathrm{T}_r$, $\mathrm{T}_{r+1}$ and $\mathrm{T}_{r+2}$ in the binomial expansion of $(a+b)^{12}$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $\left(\sqrt[4]{3} + \sqrt[3]{4}\right)^{12}$. Then $p + q$ is equal to :

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ is equal to :

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:

If the constant term in the expansion of $\left(1 + 2x - 3x^3\right)\left(\dfrac{3}{2}x^2 - \dfrac{1}{3x}\right)^9$ is p, then 108p is equal to ________.

If the constant term in the expansion of $\left(\dfrac{\sqrt[5]{3}}{x} + \dfrac{2x}{\sqrt[3]{5}}\right)^{12}$, $x \neq 0$, is $\alpha \times 2^8 \times \sqrt[5]{3}$, then $25\alpha$ is equal to :

If the second, third and fourth terms in the expansion of $(x + y)^n$ are 135, 30 and $\dfrac{10}{3}$, respectively, then $6(n^3 + x^2 + y)$ is equal to ________

Let $0 \le \mathrm{r} \le \mathrm{n}$. If $^{\mathrm{n}+1}C_{\mathrm{r}+1} : {^{\mathrm{n}}C_{\mathrm{r}}} : {^{\mathrm{n}-1}C_{\mathrm{r}-1}} = 55:35:21$, then $2\mathrm{n}+5\mathrm{r}$ is equal to :

Let $\alpha=\sum_{r=0}^{n}\left(4r^2+2r+1\right){}^{n}C_r$ and $\beta=\left(\sum_{r=0}^{n}\frac{{}^{n}C_r}{r+1}\right)+\frac{1}{n+1}$. If $140<\frac{2\alpha}{\beta}<281$, then the value of $n$ is __________.

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

The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+x^4(1+x)^{96}+\ldots+x^{54}(1+x)^{46}$ is $^{99}C_p-{}^{46}C_q$. Then a possible value of $p+q$ is :

The remainder when $428^{2024}$ is divided by 21 is ________.

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

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

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

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

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 __________.

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

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 $p^2-\alpha^2+6\alpha+2p$ equals

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

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

Let 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 $\sum_{r=0}^{n} \alpha_{r}=\beta^{n}-\gamma^{n}$, $\beta, \gamma \in \mathbb{N}$, then the value of $\beta^{2}+\gamma^{2}$ equals ________.