If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + .... + {C_n}{x^n}$, then the value of ${C_0} + 2{C_1} + 3{C_2} + .... + (n + 1){C_n}$ will be

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

The number of terms in the expansion of $(1 +x)^{101}  (1 +x^2 - x)^{100}$ in powers of $x$ is

Let $\left(1+x+2 x^{2}\right)^{20}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{40} x^{40}$ then $a _{1}+ a _{3}+ a _{5}+\ldots+ a _{37}$ is equal to

If ${(1 + x - 2{x^2})^6} = 1 + {a_1}x + {a_2}{x^2} + .... + {a_{12}}{x^{12}}$, then the expression ${a_2} + {a_4} + {a_6} + .... + {a_{12}}$ has the value

If $f(y) = 1 - (y - 1) + {(y - 1)^2} - {(y - 1)^{^3}} + ... - {(y - 1)^{17}},$ then the coefficient of $y^2$ in it is