The middle term in the expansion of ${(1 + x)^{2n}}$ is
$\frac{{1.3.5....(5n - 1)}}{{n!}}{x^n}$
$\frac{{2.4.6....2n}}{{n!}}{x^{2n + 1}}$
$\frac{{1.3.5....(2n - 1)}}{{n!}}{x^n}$
$\frac{{1.3.5....(2n - 1)}}{{n!}}{2^n}{x^n}$
If the term without $x$ in the expansion of $\left( x ^{\frac{2}{3}}+\frac{\alpha}{ x ^3}\right)^{22}$ is $7315$ , then $|\alpha|$ is equal to $...........$.
In the expansion of the following expression $1 + (1 + x) + {(1 + x)^2} + ..... + {(1 + x)^n}$ the coefficient of ${x^k}(0 \le k \le n)$ is
Coefficient of ${x^2}$ in the expansion of ${\left( {x - \frac{1}{{2x}}} \right)^8}$ is
If the third term in the binomial expansion of ${\left( {1 + {x^{{{\log }_2}\,x}}} \right)^5}$ equals $2560$, then a possible value of $x$ is