Let $\left(\frac{n}{k}\right)=\frac{n !}{k !(n-k) !}$. Then the sum $\frac{1}{2^{10}} \sum \limits_{ k =0}^{10}\left(\frac{10}{ k }\right) k ^2$, lies in the interval

For natural numbers $m,n$ ,if ${\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n} = 1 + {a_1}y + {a_2}{y^2} + \ldots \;$ and $a_1= a_2=10,$ then $(m,n)$ =______. 

If the sum of the coefficients in the expansion of ${(1 - 3x + 10{x^2})^n}$ is $a$ and if the sum of the coefficients in the expansion of ${(1 + {x^2})^n}$ is $b$, then

If ${a_r}$ is the coefficient of ${x^r}$, in the expansion of ${(1 + x + {x^2})^n}$, then ${a_1} - 2{a_2} + 3{a_3} - .... - 2n\,{a_{2n}} = $

If  the number of terms in the expansion  of ${\left( {1 - \frac{2}{x} + \frac{4}{{{x^2}}}} \right)^n},x \ne 0$ is $28$ then the sum of the coefficients of all the terms in this expansion, is :

The sum of last eight consecutive coefficients in the expansion of $(1+x)^{15}$ is