If $^{2017}C_0 + ^{2017}C_1 + ^{2017}C_2 + ...... + ^{2017}C_{1008} = \lambda^2$ where $\lambda > 0$,then the remainder when $\lambda$ is divided by $33$ is:

The sum of the series $\binom{20}{0} - \binom{20}{1} + \binom{20}{2} - \binom{20}{3} + \dots + \binom{20}{10}$ is:

If $(1 + x)^n = c_0 + c_1x + c_2x^2 + c_3x^3 + \dots + c_nx^n$,then the value of $c_0 - 3c_1 + 5c_2 - \dots + (-1)^n(2n + 1)c_n$ is

If $P_{n}$ denotes the product of the binomial coefficients in the expansion of $(1+x)^{n}$,then $\frac{P_{n+1}}{P_n}=$

If ${C_0}, {C_1}, {C_2}, ......., {C_n}$ are the binomial coefficients,then $2.{C_1} + {2^3}.{C_3} + {2^5}.{C_5} + ....$ equals

For $z \in \mathbb{C}$,if $(1+z)^n = 1 + { }^n C_1 z + { }^n C_2 z^2 + \ldots + { }^n C_n z^n$ and $\sum_{r=0}^{100} { }^{100} C_r \sin(rx) = \left(2 \cos \frac{x}{2}\right)^{100} \sin(kx)$,then $k =$