Binomial Theorem Questions (605)

The sum of coefficients of integral powers of \(x\) in the binomial expansion of \((1-2\sqrt{x})^{50}\) is

The sum of \(1 + n\left(1 - \dfrac{1}{x}\right) + \dfrac{n(n+1)}{2!}\left(1-\dfrac{1}{x}\right)^2 + \cdots \infty\) will be

If $(1+x+x^2)^n = a_0 + a_1x + a_2x^2 + ... + a_{2n}x^{2n}$, then the value of $a_1 - a_3 + a_5 - a_7 + ... $ is equal to :

Given \((2-x^2)\cdot\left((1+2x+3x^2)^6+(1-4x^2)^6\right)\), find the coefficient of \(x^2\).

The value of \(r\) for which \({}^{20}C_r\,{}^{20}C_0 + {}^{20}C_{r-1}\,{}^{20}C_1 + {}^{20}C_{r-2}\,{}^{20}C_2 + \cdots + {}^{20}C_0\,{}^{20}C_r\) is maximum, is __________.

If \(x^n - 1\) is divisible by \(x - k\), then the least positive integral value of \(k\) is __________.

Find the coefficient of \(a^3b^4c\) in the expansion of \((1 + a - b + c)^9\).

Let $a=1+\dfrac{^2C_2}{3!}+\dfrac{^3C_2}{4!}+\dfrac{^4C_2}{5!}+\cdots$ and $b=1+\dfrac{^1C_0+^1C_1}{1!}+\dfrac{^2C_0+^2C_1+^2C_2}{2!}+\dfrac{^3C_0+^3C_1+^3C_2+^3C_3}{3!}+\cdots$. Then $\dfrac{2b}{a^2}$ is equal to:

If the value of x is so small that x2 and greater powers can be neglected, then \(\frac{\sqrt{1+x}+\sqrt[3]{(1-x)^{2}}}{1+x+\sqrt{1+x}}\) is equal to

If C0, C1, C2, ..., Cn denote the binomial coefficients in the expansion of (1 + x)n, then \(\sum_\limits{r=0}^{n}\) (-1)r nCr = \(\frac{1+r \log _{e} 10}{\left(1+\log _{e} 10^{n}\right)^{r}}\) is equal to

If the $1011$th term from the end in the binomial expansion of $\left(\dfrac{4x}{5}-\dfrac{5}{2x}\right)^{2022}$ is $1024$ times the $1011$th term from the beginning, then $32|x|$ is equal to

If $\displaystyle\sum_{r=1}^{30}\dfrac{r^{2}\binom{30}{r}^{2}}{\binom{30}{r-1}}=\alpha\times 2^{29}$, then $\alpha$ is equal to \rule{2cm}{0.4pt}.

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 $2A=5B$, then $n$ is equal to:

Let the coefficients of three consecutive terms $T_{r},T_{r+1}$ and $T_{r+2}$ 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 $(\sqrt[3]{4}+\sqrt[4]{3})^{12}$. Then $p+q$ is equal to:

$\displaystyle\sum_{k=0}^{6}{}^{51-k}C_3$ is equal to:

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

If $13^n - 10^8$ is divided by 162, then the remainder is

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 the terms in this expansion, is

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

For some $n\ne 10$, 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:

If 1399 is divided by 81, the remainder is 46. The value of a is

If $\dfrac{1}{n+1}\binom{n}{n}+\dfrac{1}{n}\binom{n}{n-1}+\cdots+\dfrac{1}{2}\binom{n}{1}+\binom{n}{0}=\dfrac{1023}{10}$, then $n$ is equal to

If the coefficient of $x^7$ in $\left(ax-\dfrac{1}{bx^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax+\dfrac{1}{bx^2}\right)^{13}$ are equal, then $a^4b^4$ is equal to:

The mean of the coefficients of $x,x^2,\ldots,x^7$ in the binomial expression of $(2+x)^9$ is _________.

In the expansion of \((1 + 2x + 3x^2 + \cdots)^{-3/2}\), there is no term containing \(x^5\). Is this statement true?

Let $n$ be a positive integer and $(1 + x + x^2)^n = a_0 + a_1 x + a_2 x^2 + \ldots + a_{2n-1}x^{2n-1} + a_{2n}x^{2n}$, then:

The least value of $n$ for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7}+\sqrt[12]{11})^{n}$ is $183$, is:

Let $(a+bx+cx^2)^{10}=\displaystyle\sum_{i=0}^{20} p_i x^i$, $a,b,c\in\mathbb{N}$. If $p_1=20$ and $p_2=210$, then $2(a+b+c)$ is equal to

If in the expansion of $(1+x)^{p}(1-x)^{q}$, the coefficients of $x$ and $x^{2}$ are $1$ and $-2$ respectively, then $p^{2}+q^{2}$ 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 $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals:

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

Let the number $(22)^{2022}+(2022)^{22}$ leave the remainder $\alpha$ when divided by $3$ and $\beta$ when divided by $7$. Then $(\alpha^2+\beta^2)$ is equal to

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

If the coefficients of $x^7$ in $\left(ax^2+\dfrac{1}{2bx}\right)^{11}$ and $x^{-7}$ in $\left(ax-\dfrac{1}{3bx^2}\right)^{11}$ are equal, then

In the expansion of \(\left(\dfrac{1}{x^2} - x^3\right)^n\), \(n \in \mathbb{N}\), if the sum of the coefficients of \(x^5\) and \(x^{10}\) is 0, then \(n\) is

The sum of all rational terms in the expansion of $\bigl(1+2^{1/3}+3^{1/2}\bigr)^{6}$ is equal to \rule{2cm}{0.4pt}.

The coefficient of x4 in the expansion of (1 + x + x2 + x3)11 is

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

Let the coefficients of three consecutive terms $T_r, T_{r+1}$ and $T_{r+2}$ 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 $(\sqrt[3]{3} + \sqrt[4]{4})^{12}$. Then $p + q$ is equal to:

If the fourth term in the Binomial expansion of \(\left(\dfrac{2}{x} + x^{\log_8 x}\right)^6\) \((x > 0)\) is \(20 \times 8^7\), then a value of \(x\) is ___________.

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

If the coefficients of $x$ and $x^2$ in $(1+x)^p(1-x)^q$ are $4$ and $-5$ respectively, then $2p+3q$ is equal to

If $\alpha=1+\displaystyle\sum_{r=1}^{6}(-3)^{r-1}\binom{12}{2r-1}$, then the distance of the point $(12,\sqrt{3})$ from the line $\alpha x-\sqrt{3}\,y+1=0$ is \rule{2cm}{0.4pt}.

The value of \(\frac{1}{81^{n}}-\frac{10}{81^{n}}\) \({ }^{2 n} C_{1}+\frac{10^{2}}{81^{n}} \cdot{ }^{2 n} C_{2}+\frac{10^{3}}{81^{n}} \cdot{ }^{2 n} C_{3}\) + .... + \(\frac{10^{2 n}}{81^{n}}\) is equal to:

The least value of $n$ for which the number of integral terms in the Binomial expansion of $\left(\sqrt[3]{7} + \sqrt[12]{11}\right)^n$ is 183, is:

The coefficient of $x^7$ in $(1-x+2x^3)^{10}$ is __________.

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:

Let for $x\in\mathbb{R}$, $S_0(x)=x$, $S_k(x)=C_kx+k\int_0^x S_{k-1}(t)\,dt$ where $C_0=1$, $C_k=1-\int_0^1 S_{k-1}(x)\,dx$, $k=1,2,3,\ldots$. Then $S_2(3)+6C_3$ is equal to _______.

If $\displaystyle\sum_{r=1}^{30} \frac{r^2 \left({}^{30}C_r\right)^2}{{}^{30}C_{r-1}} = \alpha \times 2^{29}$, 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