If the sum of the coefficients of all even powers of $x$ in the product $\left(1+x+x^{2}+\ldots+x^{2 n}\right)\left(1-x+x^{2}-x^{3}+\ldots+x^{2 n}\right)$ is $61,$ then $\mathrm{n}$ is equal to

A possible value of $^{\prime}x^{\prime}$, 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:

Value $\sum\limits_{r = 0}^{15} {\left( {{}^{15}{C_r}{}^{40}{C_{15}}{}^{20}{C_r} - {}^{35}{C_{15}}{}^{15}{C_r}{}^{25}{C_r}} \right)} $ is-

If ${\left( {1 + x} \right)^n} = {c_0} + {c_1}x + {c_2}{x^2} + {c_3}{x^3} + ...... + {c_n}{x^n}$ , then the value of ${c_0} - 3{c_1} + 5{c_2} - ........ + {( - 1)^n}\,(2n + 1){c_n}$ is

If $a$ and $d$ are two complex numbers, then the sum to $(n + 1)$ terms of the following series $a{C_0} - (a + d){C_1} + (a + 2d){C_2} - ........$ is

If the expansion in powers of $x$ of the function  $\frac{1}{{\left( {1 - ax} \right)\left( {1 - bx} \right)}}$ is ${a_0} + {a_1}x + {a_2}{x^2} + \;{a_3}{x^3} + \; \ldots......$ then  ${a_n}$ is