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

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 

If $(1+x)^{n}=C_{0}+C_{1}x+C_{2}x^{2}+...+C_{n}x^{n}$, then the value of $C_{0}^{2}+\dfrac{C_{1}^{2}}{2}+\dfrac{C_{2}^{2}}{2}+.........+\dfrac{C_{n}^{2}}{n+1}$ is 

Verify that: $^{15}C_{8}+^{15}C_{9}-^{15}C_{6}-^{15}C_{7} = 0$

The total no. of six digit numbers ${ x }_{ 1 }{ x }_{ 2 }{ x }_{ 3 }{ x }_{ 4 }{ x }_{ 5 }{ x }_{ 6 }$ having the property ${ x }_{ 1 }<{ x }_{ 2 }\le { x }_{ 3 }<{ x }_{ 4 }<{ x }_{ 5 }\le { x }_{ 6 }$, is equal to