If the number of terms in the expansion of $(x - 2y + 3z)^n$ is $45$,then $n=$

The lowest integer which is greater than $\left(1+\frac{1}{10^{100}}\right)^{10^{100}}$ is $.....$

If $(2+\sqrt{3})^{49}+(\sqrt{3}-2)^{49}=a+b \sqrt{3}$,where $a, b \in \mathbb{Q}$,then

If the sum of the coefficients in the expansion of $({\alpha ^2}{x^2} - 2\alpha x + 1)^{51}$ vanishes,then the value of $\alpha$ is

The coefficient of $x^{19}$ in the polynomial $(x-1)(x-2^1)(x-2^2) \dots (x-2^{19})$ is:

Let $[x]$ denote the greatest integer less than or equal to $x$. If for $n \in N$,$(1-x+x^3)^n = \sum_{j=0}^{3n} a_j x^j$,then $\sum_{j=0}^{[\frac{3n}{2}]} a_{2j} + 4 \sum_{j=0}^{[\frac{3n-1}{2}]} a_{2j+1}$ is equal to