Binomial Theorem Questions (605)

The number of integral terms in the expansion of $\left(3^{1/2}+5^{1/4}\right)^{680}$ is equal to

If $a_\alpha$ is the greatest term in the sequence $a_n=\dfrac{n^3}{n^4+147}$, $n=1,2,3,\ldots$, then $\alpha$ is equal to ______.

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 coefficient of $x^3$ and $x^{-13}$ is equal to ______.

The coefficient of $x^{2012}$ in the expansion of $(1-x)^{2008}(1+x+x^2)^{2007}$ is equal to

Find the 7th term of \(\left(3x^2 - \dfrac{1}{3}\right)^{10}\).

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:

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

The coefficient of $x^{48}$ in $(1+x)+2(1+x)^2+3(1+x)^3+\cdots+100(1+x)^{100}$ is equal to

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

For Problems 18–20: If \((1 + x + x^2)^{20} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{40} x^{40}\), then answer the following questions.19. The value of \(a_0^2 - a_1^2 + a_2^2 - \cdots - a_{19}^2\) is

The term independent of \(x\) in expansion of \(\left(\dfrac{x+1}{x^{2/3} - x^{1/3} + 1} - \dfrac{x-1}{x - x^{1/2}}\right)^{10}\) is

In the expansion of \((1+x)^n\), if \(C_0 + 2\frac{C_1}{C_0} + 3\frac{C_2}{C_1} + \ldots + (n+1)\frac{C_n}{C_{n-1}}\) is equal to

If \(p + q = 1\), then find the value of \(\sum_{r=0}^{n} r^2 \cdot {}^n C_r p^r q^{n-r}\).

For Problems 18–20: If \((1 + x + x^2)^{20} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{40} x^{40}\), then answer the following questions.20. The value of \(a_0 + 3a_1 + 5a_2 + \cdots + 81a_{40}\) is

Consider \(f(n) = \binom{2n+1}{1}C + \binom{2n+1}{3}C\left(2^3\right) + \binom{2n+1}{5}C\left(2^6\right) + \binom{2n+1}{7}C\left(2^9\right) + \ldots + \binom{2n+1}{(2n+1)}C\left(2^{3n}\right)\)When \(f(n)\) divided by 5 then which one of the following is true about the remainder

The coefficient of a^4 b^8 c^9 d^9 in the expansion of (abc + abd + acd + bcd)^{10} is

The term independent of x in the expansion of \(\left(\frac{1}{60}-\frac{x^{8}}{81}\right) \cdot\left(2 x^{2}-\frac{3}{x^{2}}\right)^{6}\) is equal to

The value of \(\displaystyle\sum_{r=0}^{20} r(20-r)\,({}^{20}C_r)^2\) is equal to

The value of C_0 + \frac{C_1}{1 \cdot 3} + \frac{C_2}{2 \cdot 3} + \frac{C_3}{3 \cdot 3} + \cdots + \frac{C_n}{(n+1) \cdot 3} 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 binomial expansion of \(\left(\sqrt{x} - \frac{k}{x}\right)^{10}\) is 405, then \(|k|\) equals

$\frac{^nC_0}{2} + \frac{^nC_1}{6} + \frac{^nC_2}{10} + \frac{^nC_3}{14} + .... + \frac{(-1)^n{}^nC_n}{(4n+2)}, n \in \mathbb{N}$ is equal to:

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

If A and B are the coefficients of xn in the expansions of (1 + x)2n and (1 + x)2n-1 respectively, then

If $({}^{30}C_1)^2+2({}^{30}C_2)^2+3({}^{30}C_3)^2+\cdots+30({}^{30}C_{30})^2=\dfrac{\alpha\cdot60!}{(30!)^2}$, then $\alpha$ is equal to:

Let the sum of the coefficients of the first three terms in the expansion of $\left(x-\dfrac{3}{x^2}\right)^n$, $x\neq0$, $n\in\mathbb{N}$, be 376. Then the coefficient of $x^4$ is ___.

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

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

If the coefficient of $x^9$ in $\left(\alpha x^3+\dfrac{1}{\beta x}\right)^{11}$ and the coefficient of $x^{-9}$ in $\left(\alpha x-\dfrac{1}{\beta x^3}\right)^{11}$ are equal, then $(\alpha\beta)^2$ is equal to ___.

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 these three terms, is ___.

The number of dissimilar terms in the expansion of \((a + 2b + 3c)^8\) is

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

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

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

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

If the constant term in the binomial expansion of $\left(\dfrac{x^{5/2}}{2}-\dfrac{4}{x^\ell}\right)^9$ 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 ___.

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

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

Let the sixth term in the binomial expansion of $\left(\sqrt{2^{\log_2(10-3^x)}}+\sqrt[5]{2^{(x-2)\log_2 3}}\right)^m$, in the increasing powers of $2^{(x-2)\log_2 3}$, be 21. If the binomial coefficients of the second, third and fourth terms are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $x$ is ___.

Find the 7th term of \(\left(\dfrac{4x}{5} - \dfrac{5}{2x}\right)^9\) when expanded in ascending powers of \(x\).

The constant term in the expansion of \(\left(x^2 - \dfrac{3}{x}\right)^6\) is:

The coefficient of x50 in the expansion of S = (1 + x)1000 + 2x(1 + x)999 + 3x2(1 + x)998 + ... + 10001x1000 is 

Let $f_1(x) = (x-2)^2$, $f_2(x) = ((x-2)^2 - 2)^2$, $f_3(x) = \left((x-2)^2 - 2)^2 - 2\right)^2$, and so on; so that $f_k(x) = \left(...\left((x-2)^2 - 2\right)^2 - ... - 2\right)^2 = A_k + B_k x + C_k x^2 + D_k x^3 + ...$. Then $B_8$ is equal to:

Let $f_1(x) = (x-2)^2$, $f_2(x) = ((x-2)^2 - 2)^2$, $f_3(x) = \left((x-2)^2 - 2)^2 - 2\right)^2$, and so on; so that $f_k(x) = \left(...\left((x-2)^2 - 2\right)^2 - ... - 2\right)^2 = A_k + B_k x + C_k x^2 + D_k x^3 + ...$. Then $C_k$ is equal to:

If the middle term of the expression $(1 + x)^{24}, x > 0$, is the only greatest term of the expansion, then:

The sum of the series $^nC_1 + \frac{1+2}{2}{}^nC_2 + \frac{1+2+3}{3}{}^nC_3 + .... + \frac{1+2+3+...+n}{n}{}^nC_n$ is equal to:

Let $x = (5\sqrt{5} + 8)^{2n+1}$, $n \in \mathbb{N}$, then :

$^nC_0 x^{2n} + \frac{^nC_1}{2}x^{2n-2}(2-x^2) + \frac{^nC_2}{3}x^{2n-4}(2-x^2)^2 + .... + \frac{^nC_n(2-x^2)^n}{(n+1)} =$

The term independent of $x$ in $(1+x+x^{-2}+x^{-3})^{10}$ is $n$. Then the last digit of $(n+2)^n$ is

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