In the expansion of $(1 + x + y + z)^4$,the ratio of the coefficients of $x^2y$,$xy^2z$,and $xyz$ is:

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

If the coefficients of $T_r, T_{r+1}, T_{r+2}$ terms of $(1+x)^{14}$ are in $A.P.$,then $r =$

If ${a_1}, {a_2}, {a_3}, {a_4}$ are the coefficients of any four consecutive terms in the expansion of ${(1 + x)^n}$,then $\frac{{{a_1}}}{{{a_1} + {a_2}}} + \frac{{{a_3}}}{{{a_3} + {a_4}}}$ =

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