If the coefficients of the three successive terms in the binomial expansion of $(1 + x)^n$ are in the ratio $1 : 7 : 42,$ then the first of these terms in the expansion is (in $^{th}$)

$A$ possible value of $x$,for which the ninth term in the expansion of $\left\{3^{\log _{3} \sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}\right\}^{10}$ in the increasing powers of $3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}$ is equal to $180$,is:

If $n$ is even and the middle term in the expansion of $\left(x^2+\frac{1}{x}\right)^n$ is $924 x^6$,then $n$ is equal to

Find the coefficient of $a^{5} b^{7}$ in $(a-2 b)^{12}$.

Sum of the coefficients of $x^4$ and $x^6$ in the expansion of $(1+x-x^2)^6$ is

If $\frac{T_2}{T_3}$ in the expansion of $(a + b)^n$ and $\frac{T_3}{T_4}$ in the expansion of $(a + b)^{n + 3}$ are equal,then $n=$