Let $0 \leq \mathrm{r} \leq \mathrm{n}$. If ${ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}+1}:{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}:{ }^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}-1}=55: 35: 21$, then $2 n+5 r$ is equal to:

If $7^{th}$ term from beginning in the binomial expansion ${\left( {\frac{3}{{{{\left( {84} \right)}^{\frac{1}{3}}}}} + \sqrt 3 \ln \,x} \right)^9},\,x > 0$  is equal to $729$ , then possible value of $x$ is

Middle term in the expansion of ${(1 + 3x + 3{x^2} + {x^3})^6}$ is

The coefficient of $x^{4}$ is the expansion of $\left(1+\mathrm{x}+\mathrm{x}^{2}\right)^{10}$ is

The term independent of $x$ in the expansion of ${\left( {{x^2} - \frac{{3\sqrt 3 }}{{{x^3}}}} \right)^{10}}$ is

If the $6^{th}$ term in the expansion of the binomial ${\left[ {\sqrt {{2^{\log (10 - {3^x})}}} + \sqrt[5]{{{2^{(x - 2)\log 3}}}}} \right]^m}$ is equal to $21$ and it is known that the binomial coefficients of the $2^{nd}$, $3^{rd}$ and $4^{th}$ terms in the expansion represent respectively the first, third and fifth terms of an $A.P$. (the symbol log stands for logarithm to the base $10$), then $x = $