If ${ }^n \mathrm{C}_0+\frac{1}{2}{ }^n \mathrm{C}_1+\frac{1}{3}{ }^n \mathrm{C}_2$$$+\ldots \frac{1}{n}^n C_{n-1}+\frac{1}{n+1}{ }^n C_n=\frac{1023}{10} \,\,\, then \,\,\,\,\mathrm{n}=$$

The value of $$\frac{{ }^{10} \mathrm{C}_{\mathrm{r}}}{{ }^{11} \mathrm{C}_{\mathrm{r}}}$$, when both the numerator and denominator are at their greatest values, is

The difference between the maximum values of $${ }^6 C_r$$ and $${ }^n C_r$$ is 16, then $$n=$$

If $${ }^{11} \mathrm{C}_4+{ }^{11} \mathrm{C}_5+{ }^{12} \mathrm{C}_6+{ }^{13} \mathrm{C}_7={ }^{14} \mathrm{C}_5$$, then value of $$\mathrm{r}$$ is