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

If the coefficient of $a^{7} b^{8}$ in the expansion of $(a+2b+4ab)^{10}$ is $K \cdot 2^{16}$,then $K$ is equal to .... .

If $a, b, c, d$ are any four consecutive coefficients of any expanded binomial,then $\frac{a + b}{a}, \frac{b + c}{b}, \frac{c + d}{c}$ are in

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:

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