The coefficient of ${x^5}$ in the expansion of ${(x + 3)^6}$ is

If the coefficient of $x ^{15}$ in the expansion of $\left(a x^3+\frac{1}{b x^{\frac{1}{3}}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(a x^{\frac{1}{3}}-\frac{1}{b x^3}\right)^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a, b) :$

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

The coefficient of $x^{7}$ in the expression $(1+x)^{10}+x(1+x)^{9}+x^{2}(1+x)^{8}+\ldots+x^{10}$ is

The greatest value of the term independent of $x$ in the expansion of ${\left( {x\sin \theta  + \frac{{\cos \theta }}{x}} \right)^{10}}$ is

In the expansion of $(1 + x + y + z)^4$ the ratio of coefficient of $x^2y, xy^2z, xyz$ are