In the expansion of ${(1 + x)^5}$, the sum of the coefficient of the terms is

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

Let $\alpha=\sum_{\mathrm{r}=0}^{\mathrm{n}}\left(4 \mathrm{r}^2+2 \mathrm{r}+1\right)^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}$ and $\beta=\left(\sum_{\mathrm{r}=0}^{\mathrm{n}} \frac{{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}}{\mathrm{r}+1}\right)+\frac{1}{\mathrm{n}+1}$. If $140<\frac{2 \alpha}{\beta}<281$ then the value of $n$ is...............

Total number of terms in the expansion of $\left[ {{{\left( {1 + x} \right)}^{100}} + {{\left( {1 + {x^2}} \right)}^{100}}{{\left( {1 + {x^3}} \right)}^{100}}} \right]$ is

In the expansion of ${(1 + x)^{50}},$ the sum of the coefficient of odd powers of $x$ is

If $(1 + x - 3x^2)^{2145} = a_0 + a_1x + a_2x^2 + .........$ then $a_0 - a_1 + a_2 - a_3 + ..... $ ends with