Binomial Theorem

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

If the sum of the mean and the variance of a binomial distribution for 5 trials is 1.8 , then $p=$