The middle term in the expansion of ${(1 + x)^{2n}}$ is
$\frac{{1.3.5....(5n - 1)}}{{n!}}{x^n}$
$\frac{{2.4.6....2n}}{{n!}}{x^{2n + 1}}$
$\frac{{1.3.5....(2n - 1)}}{{n!}}{x^n}$
$\frac{{1.3.5....(2n - 1)}}{{n!}}{2^n}{x^n}$
The coefficients of three successive terms in the expansion of ${(1 + x)^n}$ are $165, 330$ and $462$ respectively, then the value of n will be
Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{ x }-\frac{6}{ x ^{\frac{3}{2}}}\right)^{ n }, n \leq 15$. If the sum of the coefficients of the remaining terms in the expansion is $649$ and the coefficient of $x^{-n}$ is $\lambda \alpha$, then $\lambda$ is equal to $..........$.
The number of integral terms in the expansion of $(7^{1/3} + 11^{1/9})^{6561}$ is :-
In the expansion of $(1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0$, the sum of the coefficient of $x^3$ and $x^{-13}$ is equal to
The term independent of $x$ in ${\left[ {\sqrt{\frac{ x }{3}} + \frac{{\sqrt 3 }}{{{x^2}}}} \right]^{10}}$ is