If $\displaystyle ^{n}C_{3}= ^{n}C_{5'}$ then find the value of n:

If $(1+x)^{15}=C_{0}+C_{1}xC_{2}x^{2}+...+C_{15}x^{15},$ then $^{15}C_{0}^{2}- ^{15}C_{1}^{2}+^{15}C_{2}^{2}- ^{15}C_{2}^{3}+... ^{15}C_{15}^{2}$ is equal to

The value $r$ for which $\left ( \overset{30}{r} \right )\left ( \overset{15}{r} \right )+\left ( \overset{30}{r-1} \right )\left ( \overset{15}{1} \right ) +..........+\left ( \overset{15}{r} \right )\left ( \overset{30}{r} \right )$ is maximum is/are

If ${a_n} = \sum\limits_{r = 0}^n {\frac{1}{{^n{C_r}}},}$ then ${a_n} = \sum\limits_{r = 0}^n {\frac{r}{{^n{C_r}}},}$ equals