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 :

$\frac{{{C_1}}}{{{C_0}}} + 2\frac{{{C_2}}}{{{C_1}}} + 3\frac{{{C_3}}}{{{C_2}}} + .... + 15\frac{{{C_{15}}}}{{{C_{14}}}} = $

Let n and k be positive integers such that $n \ge \frac{{k(k + 1)}}{2}$. The number of solutions $({x_1},{x_2},....{x_k})$, ${x_1} \ge 1,{x_2} \ge 2,....{x_k} \ge k,$ all integers, satisfying ${x_1} + {x_2} + .... + {x_k} = n$, is

If ${(1 + x)^{15}} = {C_0} + {C_1}x + {C_2}{x^2} + ...... + {C_{15}}{x^{15}},$ then ${C_2} + 2{C_3} + 3{C_4} + .... + 14{C_{15}} = $

The coefficient of $t^{50}$ in $(1 + t^2)^{25} (1 + t^{25}) (1 + t^{40}) (1 + t^{45}) (1 + t^{47})$ is

Statement $-1$: $\mathop \sum \limits_{r = 0}^n \left( {r + 1} \right)\left( {\begin{array}{*{20}{c}}n\\r\end{array}} \right) = \left( {n + 2} \right){2^{n - 1}}$

Statement $-2$:$\;\mathop \sum \limits_{r = 0}^n \left( {r + 1} \right)\left( {\begin{array}{*{20}{c}}n\\r\end{array}} \right){x^r}\; = {\left( {1 + x} \right)^n} + nx{\left( {1 + x} \right)^{n - 1}}$