The number of terms which are free from radical signs in the expansion of $(y^{1/5} + x^{1/10})^{55}$ is

Let the coefficients of powers of $x$ in the $2nd$,$3rd$,and $4th$ terms in the expansion of $(1+x)^{n}$,where $n$ is a positive integer,be in arithmetic progression. Then,the sum of the coefficients of odd powers of $x$ in the expansion is

The sum of all rational terms in the expansion of $(1+2^{1/3}+3^{1/2})^6$ is equal to . . . . . . .

Let the coefficients of three consecutive terms $T_r$,$T_{r+1}$,and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ be in a $G.P.$ and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $(\sqrt[4]{3}+\sqrt[3]{4})^{12}$. Then $p+q$ is equal to:

The constant term in the expansion of $\left(\frac{x}{2}+\frac{1}{x}+\sqrt{2}\right)^5$ is $\frac{a \sqrt{2}}{2}$,then $a=$

In the expansion of $(5^{1/2} + 7^{1/8})^{1024}$,the number of integral terms is