If the sum of the coefficients in the expansion of ${(x + y)^n}$ is $1024$, then the value of the greatest coefficient in the expansion is

The sum of all those terms which are rational numbers in the expansion of $\left(2^{1 / 3}+3^{1 / 4}\right)^{12}$ is:

If the coefficients of $x$ and $x^{2}$ in the expansion of $(1+x)^{p}(1-x)^{q}, p, q \leq 15$, are $-3$ and $-5$ respectively, then the coefficient of $x ^{3}$ is equal to $............$

In the binomial $(2^{1/3} + 3^{-1/3})^n$, if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is $1/6$ , then $n =$

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

The coefficient of ${x^3}$ in the expansion of ${\left( {x - \frac{1}{x}} \right)^7}$ is