Binomial Theorem Questions (605)

If $C_0, C_1, C_2, \ldots$ are the binomial coefficients where $C_r = {}^nC_r$. Let $S = C_0 + C_1 + 2C_2 + 3C_3 + \cdots + nC_n$, then $S$ is equal to

If $5^7$ is divided by $52$, then the remainder obtained is

Find $2^{3b}$

Given $\left(\sqrt[4]{3} + \sqrt[4]{2}\right)^{10}$, find the term independent of $z$.

Given $(1 + x + 2x^2)^{11} = 1 + a_1x + a_2x^2 + \ldots + a_{12}x^{12}$, find $a_1 + a_2 + \ldots + a_{12}$.

If the sum of the coefficients in the expansion of \(\left(\frac{1}{3}-15 x+15 x^{3}\right)^{2014}\)(17x - 17x5 + 3)2015 is n, then the number of dissimilar terms in the expansion of (1 + x)n is:

Let \((2x^2 + 3x + 4)^{10} = \sum_{r=0}^{20} a_r x^r\) where \(a_0, a_1, a_2, \ldots, a_{20}\) are constant, then find the value of \(\dfrac{a_7}{a_{13}}\).

Find the coefficient of $x^8$ in $(1+x+x^2+x^3)^6$

If $3^x = a_0 + a_1 + a_2 + \ldots + a_{6n}$ where $a_i$ are the terms in the binomial expansion, find $a_0 + a_2 + a_4 + \ldots + a_{6n}$

Find the coefficient of $x^2$ in $(1-x(1+x)^2)^7$

Given \((1+x)(1-x)^{10}(1+x+x^2)^9\)Find the coefficient of \(x^{18}\) in the expansion.

The value of \({}^{15}C_0^2 - {}^{15}C_1^2 + {}^{15}C_2^2 - \cdots - {}^{15}C_{15}^2\) is

The coefficient of \(x^7\) in the expansion of \([1 - x - x^2(1-x)]^6\) is:

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:

What are the coefficients of the first and the last term of (a + b)n?

The smallest natural number n, such that the coefficient of x in the expansion of \(\left(x^{2}+\frac{1}{x^{3}}\right)^{n} \text { is }^{n} C_{23}\), is

For Problems 4–6: The 2nd, 3rd, and 4th terms in the expansion of \((x + a)^n\) are 240, 720, and 1080, respectively.4. The value of \((x - a)^n\) can be

The coefficient of \(x^2\) in the expansion of the product \((2-x^2)\cdot((1+2x+3x^2)^6+(1-4x^2)^6)\) is

In the expansion of \(\left(\frac{x^a}{y}+\frac{y^b}{x}\right)^n\) there is a term independent of \(x\) and \(y\) both then a and \(b\) are related as

If Cr stands for nCr, then the sum of the series \(\frac{2\left(\frac{n}{2}\right) !\left(\frac{n}{2}\right) !}{n !}\left[C_{0}^{2}-2 C_{1}^{2}+3 C_{2}^{2}-\ldots+(-1)^{n}(n+1) C_{n}^{2}\right]\), where n is an even positive integer, is equal to

In the expansion of \(\left(2 x+\frac{1}{2 x}\right)^{6}\), if the coefficient of x2 is k times the coefficient of x-2 then k equals:

235. 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 sum: \({}^4C_1 + {}^5C_2 \cdot \dfrac{1}{2} + {}^6C_3\left(\dfrac{1}{2}\right)^2 + \cdots\) to \(\infty\).

Sum the series: \[\frac{7}{5}\left(1 + \frac{1}{10^2} + \frac{1\cdot3}{1\cdot2}\cdot\frac{1}{10^4} + \frac{1\cdot3\cdot5}{1\cdot2\cdot3}\cdot\frac{1}{10^6} + \cdots \text{ to } \infty\right).\]

Find the term independent of \(x\) in the expansion of \(\left(2x^2 - \dfrac{3}{x^3}\right)^{25}\).

Sum the series to infinite terms: \[1 + \frac{2}{6} + \frac{2\cdot5}{6\cdot12} + \frac{2\cdot5\cdot8}{6\cdot12\cdot18} + \cdots\]

Find the third term in the expansion of \((3+2x)^{3/5}\) if \(|x| > \frac{3}{2}\).

The coefficient of \(x^3\) in the expansion of \((2 - x + 3x^2)^5\) is ______.

For Problems 4–6: The 2nd, 3rd, and 4th terms in the expansion of \((x + a)^n\) are 240, 720, and 1080, respectively.5. The value of least term in the expansion 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 all the coefficients in the expansion of $(1 + x^2)^n$, then:

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

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

The largest natural number $n$ such that $3^n$ divides $66!$ is _______.

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

The coefficient of the term independent of x in the expansion of (1 + x + 2x3) \(\left(\frac{3 x^{2}}{2}-\frac{1}{3 x}\right)^{9}\) is

If$\sum$$r+1$11 11 , then$\alpha$is equal to : 10$10 -1$11$\alpha$-11 ( )$\cdot$$Cr+1 = r=0$r 10 10 10

The number of integral terms in the expansion of (5 1 1 1016$2 + 78$) is

The remainder, when $7^{103}$ is divided by $23$, is equal to:

The remainder, when $7^{103}$ is divided by 23, is equal to:

The remainder when ((64) (64) ) (64) is divided by 7 is equal to

in the expansion of ($\sqrt{2}$+ 3 1 ) ,n$\ in $N , if the ratio of 15 at term from the beginning to the 15 th term from the 3$\sqrt{3}$end is 1 6 , then the value of n$C_{3}$is:

The product of the last two digits of (1919) 1919 is

Let $\alpha,\beta,\gamma$ and $\delta$ be the coefficients of $x^{7},x^{5},x^{3}$ and $x$ respectively in the expansion of $(x+\sqrt{x^{3}-1})^{5}+(x-\sqrt{x^{3}-1})^{5},\,x>1.$ If $u$ and $v$ satisfy $\alpha u+\beta v=18$ and $\gamma u+\delta v=20$, then $u+v$ equals:

If $\displaystyle\sum_{r=0}^{5}\dfrac{\binom{11}{2r+1}}{2r+2}=\dfrac{m}{n},\ \gcd(m,n)=1,$ then $m-n$ is equal to \rule{2cm}{0.4pt}.

Remainder when $64^{32^{32}}$ is divided by 9 is equal to ______.

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

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

The sum of all possible values of $n\in\mathbf{N}$, so that the coefficients of $x$, $x^2$ and $x^3$ in the expansion of $(1+x^2)^2(1+x)^n$, are in arithmetic progression is:

The coefficient of $x^{1012}$ in the expansion of $(1+x^n+x^{253})^{10}$ where $n\leq22$ is any positive integer, is

If $\left(\dfrac{1}{{}^{15}C_0}+\dfrac{1}{{}^{15}C_1}\right)\left(\dfrac{1}{{}^{15}C_1}+\dfrac{1}{{}^{15}C_2}\right)\cdots\left(\dfrac{1}{{}^{15}C_{12}}+\dfrac{1}{{}^{15}C_{13}}\right)=\dfrac{\alpha^{13}}{{}^{14}C_0\cdot{}^{14}C_1\cdots{}^{14}C_{12}}$, then $30\alpha$ is equal to _____